Development Theory. Chapter 10

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Donald Anderson
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1 Chapter 1 Development Theory Development Theory In this section I will discuss: - further aspects of the structure and dynamics of synoptic-scale disturbances, and - derive Sutcliffe's development theory, which provides a number of practical forecastin rules.

2 In quasi-eostrophic theory the aeostrophic wind satisfies the equation: Assume f = constant Take k => k Du Dt u + fk u = a a 1 Du = k f Dt Du Dt L NM = + u O QP u h u a In the northern (southern) hemisphere, the aeostrophic wind blows to the left (riht) of the acceleration vector NH D u Dt Requirement for vertical motion w = h u a There can be no vertical motion unless there is an aeostrophic component of the wind, assumin that w = at some heiht, say at the round.

3 u Consider flow reions, where The isallobaric wind a 1 Du = k f Dt D u Dt u 1 L u a = h ρf NM p O QP u 1 = h p ρf k called the isallobaric wind Isopleths of p/ are called isallobars. Isallobaric charts, are charts on which isopleths p/ of are plotted. They are useful as forecastin aids and were particularly so before the advent of computer-produced pronostic charts. Isallobaric charts On isallobaric charts, values of p/ are normally computed from barometric tendencies for 3 hours or 4 hours In low latitudes 4 hours is more appropriate because diurnal pressure variations due to the atmospheric tide are comparable with, or larer than, typical synoptic chanes.

4 NH isallobars u a u fallin pressure risin pressure Physically, the isallobaric wind may be viewed as a crossisobaric motion in which the air accelerates or decelerates to take up the eostrophic wind velocity consistent with the new pressure field. For example, if the isobars become closer to each other locally, the air must accelerate locally as the eostrophic wind increases. To accelerate, work must be done on the air and it must therefore move with a component across the isobars towards low pressure. Note that the aeostrophic wind blows towards fallin pressure. In particular, there is aeostrophic converence towards an isallobaric low and diverence from an isallobaric hih. isallobars NOT isobars LO HI converence diverence

5 In reions where flow patterns are approximately stationary, the acceleration experienced by air parcels is represented by the advection term Du Dt u u h This tends to be the case at upper tropospheric levels where wind speeds are enerally larer than at lower levels. u a Confluence and diffluence 1 Du = k f Dt Thus u a is perpendicular to the advective acceleration. u a u u h diffluence converence confluence diffluence diverence Isobars or eopotential heiht contours illustratin patterns of confluence and diffluence and associated reions of converence and diverence (NH case).

6 Jet streams The foreoin results have application to an understandin of the circulations associated with jet streams. These are relatively narrow currents of stron winds which occur in the upper troposphere in association with planetary and synoptic scale wave disturbances. The jet stream core, the reion of stronest wind, is evident from the distribution of isotachs, which are often superimposed on upper-level charts, for example, those correspondin with the 5 mb and 5 mb levels. A typical example is the 5 mb chart shown in the next fiure: => Reional 5 mb analysis showin planetary-scale waves. Broken lines are isotachs in knots ( kn = 1 ms -1 ). Note the two jet-streaks, one just southwest of Western Australia, the other slihtly east of Tasmania.

Mean July wind field over Australia (1956-61).

7 Mean July wind field over Australia ( ). The isotach surfaces of 3, 4 and 5 ms -1 westerly wind are drawn and projections on the earth s surface of their latitude extremes are shown as dotted lines. The values in lines are the heihts in 1 s of feet (3 m) of the isotach surfaces above selected stations. y N A LO B x A' HI B' Isobaric or eopotential heiht contours (blue dashed lines), isotachs (red lines) and a typical parcel trajectory (dotted line) in a jet stream core.

8 SH. z NH cirrus cloud SH. z NH A' jet entrance A y B' jet exit B y Meridional circulation in the entrance and exit reions of a jet stream core. Arrows denote flow directions in the y - z plane. With y pointin polewards, the direction of flow alon the jet is out of the screen for the northern hemisphere and into the screen for the southern hemisphere. Typically, the air miht accelerate from ms 1 to 6 ms 1 in travellin 15 km throuh the jet entrance reion: => fv u ~ u => va ~ ~ 1ms 4 x 1 o a Assume that the air 5 km to either side of the jet is undisturbed and that the velocity maximum occurs km below the tropopause. Then w A calculation va = => w ~ 1 ms = 4 cms y The vertical displacement of a particle is then about 1.5 km, enouh to cause condensation in air equatorwards of the axis and to clear any existin cloud on the poleward side.

9 Some notes The cirrus cloud found ahead of the warm front of an extratropical cyclone is usually such jet cloud. It is formed over the cold front and blown around the upperlevel trouh to appear ahead of the warm front. Note that implicit in the foreoin calculation is the assumption that u (~ u ) can be computed eostrophically whereas v cannot be. This is consistent with scalin analyses which show that the eostrophic approximation holds in the direction across the jet, but not alon it. The same is true also of a front. Dines compensation unit area p s = surface pressure = weiht of column upper-level diverence isallobaric converence Lare-scale motions in the atmosphere are in close hydrostatic balance => The pressure at the base of a fixed column of air is proportional to the mass of air in that column; if the total mass decreases, so will the surface pressure, and vice versa.

10 Some notes In a deepenin low, the isallobaric wind, and to a lesser extent surface friction, will contribute to low-level converence. If there were no compensatin upper-level diverence, the surface pressure would rise as mass accumulated in a column - clearly a contradiction! Dines showed that low-level converence is very nearly equal to the diverence at upper levels. He pointed out that upper diverence must exceed the low-level converence when a low deepens. Because the interated diverence is a small residual of much larer, but opposin contributions at different levels, it is not practical to predict surface pressure chanes by computin the interated diverence. Note p z ρdz = p z z ( ρ ρ dz = = u)dz, z z usin the full continuity equation: The surface pressure is iven by ρ + ( ρ )= u p s = ( ρu)dz, z This is the surface pressure tendency equation. In practice, it is of little use for prediction since observations are not accurate enouh to reliably compute the riht-hand-side.

11 p s = ρu ( )dz for strict eostrophic motion the surface pressure cannot chane! strict eostrophic motion => fρu = k h p f constant => h ( ρu) = This is consistent with the fact that the eostrophic wind blows parallel with the isobars. => any local chane in surface pressure is associated entirely with aeostrophic motion. => important consequences for the movement of disturbances characterized by their surface pressure distribution, e.. cyclones and anticyclones. Sutcliffe's development theory Sutcliffe, an Enlish meteoroloist, computed the relative diverence between an upper and lower level in the troposphere usin the vorticity equation. From this he deduced the distribution of vertical motion. A knowlede of the vertical motion at a low level can be used toether with the vorticity equation at the surface to study the development of surface pressure systems such as middle latitude cyclones and anticyclones. Sutcliffe's theory has proved extremely valuable in the practice of weather forecastin.

12 The essence of Sutcliffe's theory If one can compute the difference between the fields of aeostrophic wind at a low level and at a middle or hih tropospheric level, one can deduce the horizontal diverence, and hence the field of vertical motion which must exist between these levels. Then one can locate the reions of fallin surface pressure (over risin air) and of risin surface pressure (over subsidin air). Extratropical cyclone Anticyclone Complex disturbance mb ω 4 mb ascent 6 mb 8 mb conv 1 mb conv div div conv ω ascent subsidence ω conv subsidence div conv div conv div The vertical distribution of horizontal diverence and vertical motion (characterized by ω = Dp/Dt) for an extra-tropical cyclone, an anticyclone, and for example of a more complex disturbance.

13 cyclone z H w/ < anticyclone z H w/ > 1 1 H H w/ > w w/ < w It follows that for an extra-tropical cyclone, w/ < throuhout the depth of the troposphere, and for an anticyclone w/ >. Mathematical derivation I will use (x,y,z) rather than (x,y,p) coordinates and adopt a modern approach, usin quasi-eostrophic theory. For an extra-tropical cyclone, w/ < throuhout the depth of the troposphere, and for an anticyclone w/ >. Now w/ may be obtained from the vorticity equation: w f = ( f) z + u ζ + f w ζ ζ u f = f f f ( f) z t + + ζ + z u The plan is to obtain a dianostic formula for w/ by eliminatin the time derivative on the riht hand side.

14 Finally L NM f O ζ QP = h L N M f O ψ usin QP ζ = ψ b = 1 p h usin = b and ψ t ρ ( u ) = h b Nw usin ζ f = N hw h( b). u = p ρ f / + u h b + Nw= u ζ h( u b) = f ζ + fu + Λ, Λ= f E F F E u v v u, E =, F = + QP x y x y L NM O ζ f = N hw h( b). u u ζ h( u b) = f ζ+ fu + Λ, The quantity Λ is related to the deformation of the flow and can be shown to be small, except in active frontoenetic reions. Nelectin Λ, substitution w u f = N hw+ f (ζ + f) Sutcliffe nelects the adiabatic buoyancy tendency 1 and interates between the surface and. H N h w

15 w u f = N hw+ f (ζ+ f) 1 H z dz approximately the level of the 5 mb surface 1 w w H u 1 H s f f = (ζ+ f)dz Small ~ level of nondiverence The surface vorticity tendency the surface diverence ζ s w = us ( ζ s + f) + f s 1 H u = us ζs us f (ζ+ f)dz. 1 ζ H u s = us ζ s us ζ+ Assume a linear vertical shear f ( f)dz. u uh/ us z =, u = uh/ us, ζ = ζs + ζ H/ H z 1 H u z ( ζs + 4ζ + s H H f ) dz = u ( ζ ) + u ζ + u / f ζ s = u + u ) ζ u ζ u f ( s s H/ ζ s An equation for the surface vorticity tendency 1 = ps => The surface pressure tendency can be dianosed ρ f *

16 Interpretation For a wave-like disturbance, p s is proportional to p s so that an increase in cyclonic vorticity corresponds with a lowerin of the surface pressure. There are various ways of interpretin this equation. I will discuss the mathematical way. The equation has the form where D ζs H f Dt = u ζ u / D + ( us + u ). Dt D ζs H f Dt = u ζ u / where D + ( us + u ). Dt If the thermal vorticity advection u' ζ' and the planetary vorticity tendency u H/ f are both zero, ζ s will be conserved for points movin with velocity u s +u'. => The lines defined by dx/dt = u s +u' are characteristics of the equation. In the eneral case, ζ s chanes at the rate u' ζ' u H/ f followin a characteristic.

17 The implications of D ζs H f Dt = u ζ u / are: (i) At the centre of a surface low, u s << u' => the low pressure centre will propaate in the direction of the thermal wind, or equivalently the 5 mb wind, with speed proportional to the thermal wind; This is the thermal steerin principle. It turns out that the constant of proportionality (i.e., ) is too lare and that a value of unity is more appropriate: see later => (ii) u H/ f is positive, leadin to cyclonic development for an equatorward wind and anticyclonic development for a poleward wind; this term is relatively small. (iii) The thermal vorticity advection u' ζ' is the principal contribution to the intensification or decay of systems. Consider a wave pattern in the thickness isopleths shown in the next fiure:

18 ζ' max ζ' min ζ' max LO cold Equator u' u' LO cold div div HI warm HI warm conv thickness isopleths conv A positive vorticity advection maximum, or "PVA max" Favorable for: cycloenesis ( u' ζ') max anticycloenesis ( u' ζ') min Some notes It follows from (i) and (iii) that the propaation and chane in intensity of surface depressions can be juded from a thickness chart with a superimposed surface chart. It is often the case that in most locations, u >> u s, => u 5mb u => The thermal wind contribution to the flow at 5 mb mostly dominates that due to the surface wind. This is why the thickness isopleths and the isopleths of eopotential at 5 mb have broadly similar features. It means also that the 5 mb isopleths can be used instead of the thickness charts to ive an indication as to how systems will be steered and whether or not they will row or decay.

19 Limitations: Sutcliffe's theory hihlihts the fact that a surface depression, or trouh, and an anticyclone, or ride, will tend to be displaced in the direction of shear by a process of development as distinct from translation: => There is associated diverence and converence. Used with care, it can be a useful uide in weather forecastin, but it suffers the followin limitations: (i) it nelects the effects of adiabatic heatin and coolin, represented by the term Ν h w, and diabatic effects. These effects may make important contributions to the overall flow evolution. The omission of Ν h w The omission of Ν h w overestimates the propaation speed by a factor of two. For a wavelike disturbance with total horizontal wavenumber κ, h w is proportional to κ w. Since f w / f w / H f w / + N w ( f / H + N κ ) w h But for quasi-eostrophic disturbances, f / κ N H ~ 1 Both terms are important

20 Sutcliffe s theory is a dianostic one! (ii) the theory is a dianostic one; it ives an indication of the tendencies at a iven instant or, in practical terms, for a few hours or so. The thermal field, and hence the thermal vorticity advection u' ζ' will evolve as the surface flow develops and a more complete solution, such as that provided by numerical interation, must allow this interaction to occur. We can understand this interaction from the next fiure: cold advection 1-5 mb thickness N LO mslp warm advection

21 Some notes These considerations show also a further role of the N w term in the thermodynamic equation which Sutcliffe inored. As we have seen from baroclinic instability theory, the structure of the rowin Eady wave is such that, poleward motion is associated with ascent and coolin; equatorwards motion with subsidence and warmin. In a rowin baroclinic wave, the pairs of quantities v and w and v and σ are neatively correlated in the southern hemisphere while w and s are positively correlated. Accordinly, the N w term in the thermal tendency equation opposes the horizontal temperature advection and hence the rate of increase of u' ζ'. Other points On account of the couplin between the surface and upperlevel flow, we would not expect Sutcliffe's theory to be a substitute for a ood numerical weather prediction model. (iii) The theory does not work well when u is small; i.e., in the case of "cut-off" lows. These are low pressure systems which develop in or mirate to a reion where the upper-level steerin winds are relatively liht. (iv) The theory nelects the effects of moist processes.

22 Start from f w The omea equation = L N M f ζ O QP + f L u ζ f ζ f NM QP + u ( + ) O ζ f = N hw h( b). u u ζ h( u b) = f ζ + fu + Λ, w u N hw + f = f f ( ζ + ) + Λ A dianostic equation for the w which does not rely on computin the horizontal diverence from wind observations. Called the quasi-eostrophic form of the 'ω-equation' after its counterpart in pressure coordinates. The End

Conservation of absolute vorticity. MET 171A: Barotropic Midlatitude Waves. Draw a picture of planetary vorticity

Conservation of absolute vorticity : Barotropic Midlatitude Waves Recall the important terms in the vorticity equation in the middle troposphere, near the level of non-diverence Lecture Outline 1. Conservation