Use of dynamical concepts in weather forecasting

Transcription

1 Use of dynamical concepts in weather forecasting Meteorol. Appl. 4, (1997) E B Carroll, Meteorological Office, London Road, Bracknell, Berkshire RG12 2SZ, UK A divergence-based procedure for diagnosing development, based on a two-layer model of the atmosphere, is discussed. It is pointed out that thinking in terms of vorticity advection and thermal advection is directly equivalent to considering divergence due to advective and isallobaric components of the ageostrophic wind. It is shown that such an approach is similar in basis to the quasi-geostrophic omega equation, but it is argued that it is more suitable for subjective application. However, simply estimating vertical velocity or instantaneous pressure tendency from synoptic charts is of questionable accuracy and little operational use, especially since such diagnostics are much more reliably obtained from NWP models. It is necessary to think in terms of the development and movement of features in the flow of the upper as well as the lower troposphere, and their vertical interactions. 1. Introduction Development theory is only of real, practical benefit in forecasting when it is used to construct conceptual models involving interactions and feedback between structures at different levels in the atmosphere, and their time evolution. Such models, and the dynamical reasoning on which they are based, are necessary if the forecaster is to be more than a helpless observer of the unfolding atmospheric evolution and its simulated model counterparts. When it is suspected that the numerical model evolution might be going awry (and we will not consider here how early warning of this might be received, but it may be from satellite imagery or conventional observations), it is not sufficient to theorise on the likely instantaneous effect of the inferred error on vertical velocity. It is necessary to attempt to project that error forward in time, for instance in terms of how the lowlevel flow, and thus the thermal advection, will be altered from that forecast by the numerical model, and in turn how this might affect the upper flow, from where changes will feed back into the surface development, and so on. (Some examples of such models, for cyclogenesis, trough disruption and downstream development, are presented in section 6.) Of course, even when the meteorologist attempts to do this, it is only in an approximate, qualitative way, but it can lead to important insight into the critical aspects of a forecast evolution, the scope for error in the weather forecast, and some estimate of the nature of the likely error. What tools are available for the construction of dynamical conceptual models? Since the 1960s, when quasigeostrophic (QG) theory provided the equations for numerical models, there has been a tendency amongst theoreticians to promote the QG omega equation, in any of its several forms, as the way of diagnosing vertical velocity, and thence development. However, from as far back as the 1930s, operational forecasters have used various methods based on a simple two layer model of the atmosphere, which has the atmosphere divided into two layers of divergence of opposite sign, separated by a mid-level of non-divergence where vertical velocity is at a maximum. This simple model is valid for regions with strong vertical velocity through the troposphere and marked temporal changes in surface pressure. One of the purposes of this article is to point out that divergence-based approaches are not so different after all from QG theory, and that, while they are not suitable for quantitative evaluation (as the QG omega equation is), they arguably lend themselves better to subjective application. In more recent years, potential vorticity thinking has been gaining acceptance as a way of understanding development. This powerful way of looking at the atmospheric flow has yet to achieve common usage amongst forecasters, and will not be dealt with here; for a review, see Hoskins (1997). 2. Ageostrophic motion The ageostrophic wind is that part of the wind not in geostrophic balance, i.e. the wind associated with an imbalance between Coriolis force (acting to the right of the motion in the northern hemisphere and proportional to the wind speed) and pressure gradient force (acting down the pressure gradient). By identifying the ageostrophic wind, we isolate to a good approximation the component of the horizontal wind field associated with the divergence, because the geostrophic wind has only a very small divergence. 345

2 E B Carroll Neglecting friction, acceleration in the flow is indicative of a departure from geostrophic balance: the consequent ageostrophic motion has magnitude proportional to the acceleration of the flow, and is directed to the left of the acceleration vector in the northern hemisphere. To assess it, we need the total time derivative of the wind velocity, i.e. the acceleration following the motion. However, this is difficult to gauge, and it is convenient to split it into local and advective components, giving local (i.e. isallobaric) and advective (i.e. downwind) changes in velocity respectively, and to approximate the real wind with the geostrophic wind. The downwind effects can be further subdivided into along-flow acceleration and curvature (or across-flow acceleration) effects. V V ag Therefore : ag 1 dv 1 dv = k k f dt f dt 1 V g k + V g V g f t isallobaric downwind diffluence or curvature confluence 2.1. Downwind effects term 1 term 2 Figure 1 shows development areas deduced from consideration of upper-level divergence of the ageostrophic wind due to term 2. In simple terms, the increasing pressure gradient force experienced by a parcel of air entering the westerly jet dominates over the Coriolis force which, being proportional to the wind velocity, g lags behind until the velocity increases. At the jet exit, the pressure gradient force decreases, whilst the Coriolis force takes time to decrease to geostrophic balance. The result is a geostrophic departure down the pressure gradient at the jet entrance and up the pressure gradient at the exit, and the accompanying divergence pattern and associated vertical circulations shown in Figure 1. (The cross-contour flow could equally be explained in terms of a compensating reduction in the potential energy (geopotential) of the parcel at the jet entrance as kinetic energy increases, with the opposite exchanges at the exit.) The omega equation advocate would argue that we can tell nothing about the vertical velocity without knowledge of the thermal pattern. Strictly speaking, this is true, since, by ignoring term 1, the isallobaric term, we are not considering the total ageostrophic motion. On the other hand, by observing that the wind blows through such upper-level features at a speed typically much in excess of the speed of progression of the features themselves, we can see that term 1 is relatively unimportant at upper levels in this situation. For the case of divergence due to changes in curvature along the flow (Figure 2), a similar caveat applies; it is the degree to which the wind is blowing through the feature that is important again, we are ignoring changes in the flow pattern for the moment Isallobaric effects Isallobaric effects are associated with local changes in the pressure gradient force; the attendant ageostrophic motion is directed down the isallobaric gradient. At the surface, isallobaric flow is directed into regions of pressure falls and away from regions of pressure rises (Figure 1). At upper levels, the change in geopotential height (gph) is given by the change in the 1000 hpa gph (closely related to the surface pressure tendency) plus the change in thickness, or average temperature, between 1000 hpa and the upper pressure level. The latter is much the dominant effect, with the most significant cause of isallobaric motion being horizontal variation in thermal advection. Warm advection tends to raise upper-level gph; this results in isallobaric motion Figure 1. Ageostrophic motion and vertical velocity (heavy arrows) associated with a westerly jet. Ageostrophic motion at upper levels is associated with downwind acceleration, that at lower levels with isallobaric effects. Also shown are relative strengths of pressure gradient and Coriolis forces (PGF and CF on light arrows) and development areas at the surface (A = anticyclonic, C = cyclonic). 346 Figure 2. Ageostrophic motion (arrows) and surface development areas associated with the flow of air through a fixed upper trough ridge flow pattern. (The ageostrophic motion is simply a consequence of the subgeostrophic flow in the troughs and the supergeostrophic flow in the ridges.) (A = anticyclonic, C = cyclonic.)

3 Use of dynamical concepts in weather forecasting directed away from the warm advection maximum, divergence aloft and ascent through the atmospheric column (Figure 3(a)). Cold advection has the opposite affect, with upper-level convergence and descent (Figure 3(b)). On the edges of advection areas, the opposite sign of divergence tends to occur. For instance, cold advection behind a cold front contributes to ascent at the front. During the process of frontogenesis, parcels of air are being accelerated, as at a jet entrance, leading to isallobaric motion aloft from warm to cold air (Figure 3(c)). For this reason cloud and rain are often confined to the warm side of the jet, with clear air on the cold side. The contribution of surface pressure tendency is generally that of a brake on development, since, ignoring any thickness change, it leads to equal flow into the atmospheric column at all levels if surface pressure is falling, and out of it if surface pressure is rising. The adiabatic change in the temperature of the column through ascent and descent also has a braking effect; the cooling on ascent, for instance, itself contributes a lowering of the upper tropospheric pressure level and ageostrophic motion into the region of ascent. Release of latent heat in regions of moist ascent amplifies the vertical motion, but doesn t cause ascent in the absence of any pre-existing forcing. 3. Vorticity thinking A simplified vorticity equation offers an alternative way of diagnosing divergence without reference to ageostrophic motion. It shows the fractional rate of change of absolute vorticity, ζ, following the motion to be proportional to the convergence (ignoring frictional effects and the tilting of horizontal vorticity). This equation is often visualised as the ice-skater effect, whereby pirouetting ice skaters increase their vorticity (or angular velocity) by drawing their arms in (convergence) and decrease it by stretching their arms out (divergence). Splitting the total rate of change of absolute vorticity into local and advective components shows that divergence is associated with the advection of vorticity and the local rate of change of vorticity Vorticity advection 1 dζ div H ζ dt V 1 ζ div H V + V ζ ζ t convergence local rate vorticity of change advection of vorticity Positive (negative) vorticity advection contributes towards divergence (convergence). It is easily shown that vorticity advection is proportional to the divergence diagnosed by consideration of ageostrophic motion due to downwind acceleration. Advection of shear vorticity gives divergence due to ageostrophic motion resulting from along-flow acceleration/deceleration (Figure 1), whilst advection of curvature vorticity gives divergence due to ageostrophic motion resulting from curvature (Figure 2). Figure 3. Cross-sectional views showing upper tropospheric pressure surfaces and direction of instantaneous change in height (dotted arrows). The resulting departure from geostrophic balance and induced vertical motion is shown by solid arrows. (a) warm advection, (b) cold advection, (c) jet entrance (following motion) or frontogenesis, (d) jet exit (following motion) or frontolysis. 347

4 E B Carroll Hereafter PVA and NVA will be used to refer to positive vorticity advection and negative vorticity advection Local rate of change of vorticity Such changes have their direct equivalent in the isallobaric effects discussed in section 2. The change in upperlevel vorticity is given by the change in 1000 hpa vorticity plus the change in thermal vorticity. Pursuing a line corresponding to that followed in section 2.2, it is evident that the 1000 hpa vorticity tendency contributes an equal divergence at all pressure levels, this contribution acting to make surface pressure rise (fall) where surface vorticity is increasing (decreasing), so tending to curb development. The change in thermal vorticity is, to a first approximation, given by the thermal advection pattern, with vorticity reduction (increase) over a maximum of warm (cold) advection, and trends of opposite sign on the periphery of the region of thermal advection. Again, as mentioned in section 2, adiabatic temperature changes in the column due to vertical motion tend to offset the vorticity changes, acting as a brake on the process, whilst latent heat release in regions of ascent amplifies generation of upper-level anticyclonic vorticity, and therefore enhances divergence. 4. Comparison with the QG omega equation The vorticity view of development put forward in the previous section has similarities with the traditional form of the omega equation this is not surprising, since there is nothing fundamentally different in what has gone into them. In fact, if you vertically integrate the omega equation (see section 1 of Pedder, 1997) and consider first the right-hand-side forcing term, you can begin to see how we get vorticity advection rather than its vertical derivative, and the Laplacian of the thickness advection rather than the Laplacian of the temperature advection at one level. Considering the terms in w on the left-hand side, rather than a term in 2 w/ z 2, we have w/ z, which is equal to the convergence, or negative divergence. The integrated 2 w term represents the contribution of adiabatic temperature change in the column referred to in section 2.2. Whilst significant in a quantitative sense, it does not tend to change the qualitative distribution of vertical velocity. Of the two approaches, the omega equation is undoubtedly the right one if you want to compute vertical velocity. The Laplacian form of the left-hand side allows you to solve for omega, using a computer, and assuming values of omega round the edges of the domain. It can be conveniently specified as zero, to a first approximation, at the upper and lower boundaries, and even if it is also assumed to be zero at the lateral boundaries you will not have to go far into the domain before any inaccuracies arising from this become insignificant. The mathematical form of the equation and the nature of the boundary conditions can also 348 reveal interesting things about the relationship between forcing for vertical velocity and the response, embodying notions of action at a distance and how the influence of forcing is spread horizontally and vertically. With assumed sinusoidal distributions of forcing and response (Figure 4(a)), subjective estimation of the response is straightforward, since sine functions have the property that the second derivatives are proportional to the functions themselves (but have opposite sign). However, small departures from the idealised shape can easily throw the local relationship between forcing and response into doubt, as shown in Figure 4(b), making it possible to get completely the wrong signal from a local assessment of omega forcing. In contrast, the divergence, which can be visualised as depending on the slope of the vertical-velocity profile, is much more reliable; with even the noisy example shown the divergence only changes sign once, across the mid-tropospheric vertical velocity maximum. In fact, it can be shown that the relationship between omega forcing and omega itself becomes more and more reliable as you average over ever larger volumes. This lends support to the traditional divergence-based approach to diagnosing development. 5. Sutcliffe s equation A practical drawback of the development theory so far examined is the well known potential for cancellation between terms. This stems from the approach whereby, to get the total rate of change of velocity, or of vorticity, we add the advective to the local rates of change. If the parcel of air is being advected at the same speed as the flow pattern, the two will cancel each other out, and it is not surprising that this often happens to some degree, since the pattern does get advected along to an extent. In reality, of course, there is nothing fundamentally different about isallobaric and advective accelerations that we resort to them is simply a matter of convenience. Sutcliffe (1947) took the two-layer model as a starting point, and set about finding a rule to deduce low-level, Figure 4. Schematic profiles of vertical velocity in regions of ascent, together with the sign of - 2 w/ z 2 ( omega forcing) and the sign of - w/ z (divergence) in brackets. (a) Sinusoidal profile. (b) Irregular profile.

5 Use of dynamical concepts in weather forecasting rather than upper-level, divergence, assuming the existence of a level of negligible divergence at 500 hpa. Furthermore, he considered vorticity changes following the motion, thereby circumventing the cancellation difficulty. The equation he derived had, as its so-called development term, the thermal vorticity advection (i.e. the advection of the vorticity in the hpa thickness contours by the hpa thermal wind). This was to be interpreted as an indicator of the midlevel vertical velocity and the associated generation of low-level vorticity. In addition, there was a steering term, which had the low-level vorticity steered by the thermal wind. However, although initially avoiding cancellation, Sutcliffe s final rendering of his equation reintroduced it, since his steering term always acts against his development term (Carroll, 1995). Utilising an intermediate result of Sutcliffe s, and avoiding the built-in cancellation between steering and development terms, we get the advection of the 500 hpa vorticity by the hpa thermal wind as the development term, with a steering-level wind equal to that at about 700 hpa. Hoskins et al. (1978), and, independently, Trenberth (1978), gave us an equivalent form of the omega equation (equation (A22) of Hoskins, 1997), which Hoskins called the Sutcliffe form. Again, we have a version of the omega equation giving us the three dimensional Laplacian of the vertical velocity, and a divergencediagnosing equivalent based on a vertically integrated view. For reasons given in section 4, it is suggested that the latter is the more suitable for subjective interpretation, and can often give a clear signal when the thermal and vorticity advections are acting against each other. It should be noted that there is an inherent approximation in the Sutcliffe approach whereby it ignores vertical velocity forced by frontogenesis and frontolysis (as depicted in Figures 3(c) and (d)). However, this neglect is not judged to be a serious shortcoming for the diagnosis of mid-tropospheric vertical velocity. 6. Upper and lower features development and mutual interaction In this section, the effects of thermal and vorticity advections on upper, rather than lower, tropospheric features are considered. Three examples of extra-tropical cyclogenesis archetypes are then given in terms of the concepts of thermal advection and vorticity advection. For each, it would be possible to give an equivalent account in terms of potential vorticity (PV) thinking, whereby advection of low-level thermal fields and upper-level PV features are important Thermal advection Thermal advection tends to contribute opposite vorticity and gph tendencies at lower and upper levels, e.g. warm advection is often associated with convergence, increasing vorticity and falling pressure at the surface, but divergence with decreasing vorticity and rising gph at 300 hpa it is also possible to rationalise this change in upper-level gph in terms of increasing thickness, which more than compensates for the falling 1000 hpa height. At some mid-level, the level of non-divergence (on average hpa), it has a smaller, indeterminate effect. In fact, it seems that the level of non-divergence is somewhat lower, say at about 700 hpa, in regions of strong thermal advection, rising to say 400 hpa where upper-level vorticity advection dominates. Adiabatic and diabatic heating/cooling can act to offset advective changes, e.g. cold advection to the west of the UK in winter can be significantly offset by diabatic warming through convection and adiabatic warming through descent, limiting the amount of cooling in the column and thus the fall of upper-level gph Vorticity advection The advective tendency of vorticity is an important factor in determining its local rate of change. For example, a diffluent upper trough in the westerly flow has shear vorticity and planetary vorticity advected into its axis, reinforcing it. Such troughs, therefore, tend to extend and sharpen. Planetary scale troughs and ridges tend to be slow-moving or stationary under a balance between curvature and planetary vorticity advection, whereas the shorter wavelength features associated with travelling weather systems move through the pattern as their relative vorticity is advected. (Because of the cancellation between relative vorticity advection and planetary vorticity advection associated with the planetary-scale features, the usual configuration of A and C areas does not apply.) At the nominal level of non-divergence, vorticity is, to a first approximation, conserved following the motion. For this reason vorticity features are advected around and often preserve their identities longer on a 500 hpa chart than on a 250 hpa chart. Table 1 summarises and compares the factors leading to Table 1. Factors leading to changes in geopotential height near the surface and in the upper troposphere Development Surface C area Surface A area Upper-level C area Upper-level A area Factors favouring PVA + centre of warm advection or where cold advection area borders on region of little thermal advection. NVA + centre of cold advection or where warm advection area borders on region of little thermal advection. PVA + centre of cold advection or where warm advection area borders on region of little thermal advection. NVA + centre of warm advection or where cold advection area borders on region of little thermal advection. Note: PVA and NVA refer to the upper-tropospheric Positive and Negative Vorticity Advection. 349

6 E B Carroll Figure 5. Self-development initiated by the engaging of a low-level baroclinic zone by an upper-level PVA region. Figure 6. Schematic view of trough disruption. Upper-level jets are marked by heavy arrows, low-level circulation by light arrows. Figure 7. Trough disruption shown in a 300 hpa gph forecast sequence. X marks the position of surface depression. Dashed lines are trough axes. 350

7 Use of dynamical concepts in weather forecasting changes in gph near the surface and in the upper troposphere. In any instance, thermal advection may be acting to offset vorticity advection Three examples of extra-tropical development types (a) Cyclogenesis A PVA area in a barotropic region progresses downstream without developing; development and amplification of surface and upper features will generally not take place unless a zone of surface baroclinicity is engaged, allowing constructive interaction between the two via positive feedback in the following way (Figure 5): (i) (ii) Falling pressure at a surface front ahead of an upper trough brings warm and cold advection into play. Cold advection reinforces the upper trough, whilst warm advection helps build the downstream upper ridge. The intensification of the upper trough ridge system increases the PVA, which further deepens the surface low, which increases the strength of the thermal advection, which further reinforces the upper wave, and so on. (iii) In the standard sequence of events, the occlusion process removes the PVA and low centre from the baroclinic zone, reducing thermal advection strength. As the flow becomes more barotropic, the closed circulation extends to higher levels in the atmosphere, reducing vorticity advection here and favouring a vertical rather than a tilting structure. However, if baroclinicity can be maintained in the vicinity of the depression, e.g. by injection of fresh polar air, development can continue. Since thermal advection by the lower tropospheric circulation is very important to the self-development process, the tighter the pre-existing low-level thermal gradient, the greater the potential for cyclonic development. The absolute humidity of the warm air mass is also important, since air of high water vapour content releases much latent heat on ascent, increasing upperlevel divergence. Self-development is not always initiated from aloft; the formation of a low-level perturbation can trigger the process. (b) Trough disruption Trough disruption is said to occur when the poleward portion of an upper trough becomes separated from the equatorward part and moves on, often having left a cutoff circulation (Figures 6 and 7). (i) (ii) This evolution sometimes follows the extension of a trough, especially if sufficient positive vorticity, both planetary and shear, is advected into the base of the trough, or if a cut-off region of cold air due to strong advection of cold low-level air forms in the region of the trough. As the higher latitude part of the upper trough is advected away eastwards, the low-level cyclonic circulation reinforces the lower latitude portion by in drawing cold air at low levels. Warm advection on the eastern and northern quadrants of the surface low then erodes the increasingly tenuous link between the two parts of the upper trough. (iii) Surface pressure rises across the neck of the disrupting trough, often quite rapidly, under the influence of the NVA area on the upstream side of the high-latitude portion of the trough. The disruption process is finely balanced, and it is not uncommon for small initial errors in NWP models to grow significantly and lead to poor forecasts in such situations. (c) Downstream development Weather systems can be influenced by events upstream on a time scale shorter than that in which purely advective processes can account for their development. A typical process is now described (Figure 8): Figure 8. Schematic view of downstream development. Upper-level jets are marked by heavy arrows, low-level circulation by light arrows. 351

8 E B Carroll Figure 9. Development in the northeast Atlantic downstream of strong warm advection in the northwest Atlantic. Dark lines: 300 hpa gph every 6 dam. Lighter lines, MSLP every 4 hpa. (i) (ii) A vigorous surface depression generates much warm advection ahead of it, which builds an upper ridge. The consequent veering of the thermal gradient downstream of the ridge axis leads to a tightening of that gradient and therefore a strengthening of the flow. (iii) The resulting downstream diffluent trough extends and sharpens through advection of cyclonic shear vorticity into its axis, and gives rise to a new cyclonic surface development. (iv) The intensification of this new surface feature induces cold advection on its western flank, reinforcing the upper trough. Such sequences indicate that details of the initial state of the atmosphere well upstream of the area of interest can have a bearing on the forecast within a shorter time period than might be expected. For instance, the intensity of the depression near Canada in Figure 9 could have had a bearing on the formation of the depression near the UK within the 21 hour period. References Carroll, E. B. (1995). Practical subjective application of the omega equation and Sutcliffe development theory. Meteorol. Appl., 2: Hoskins, B. J. (1997). A potential vorticity view of synoptic development. Meteorol. Appl., 4: Hoskins, B. J., Draghici, I. & Davies, H. C. (1978). A new look at the w-equation. Q. J. R. Meteorol. Soc., 104: Pedder, M. (1997). The omega equation: Q G interpretations of simple circulation features. Meteorol. Appl., 4: Sutcliffe, R. C. (1947). A contribution to the problem of development. Q. J. R. Meteorol. Soc., 73: Trenberth, K. E. (1978). On the interpretation of the diagnostic quasi-geostrophic omega equation. Mon. Wea. Rev., 106: