the barotropic forecast

4

The barotropic forecast

Before attending to the complexities of the actual atmosphere . . . it maybe well to exhibit the working of a much simplified case. (WPNP, p. 4)

To clarify the essential steps requird to obtain a numerical solution of the equationsof motion, Richardson included in his book an introductory example in which heintegrated a system equivalent to the linearized shallow water equations (WPNP,Ch. 2). He used an idealized initial pressure field defined by a simple analyticalformula, with winds derived from it by means of the geostrophic relationship. Hethen calculated the initial tendencies in the vicinity of England. On the basis of hisresults, he drew conclusions about the limitations of geostrophic winds as initialconditions. In this chapter we will re-examine his forecast using a global barotropicmodel, and reconsider his conclusions.

Richardson’s step-by-step description of his calculations in Chapter 2 of WPNPis clear and explicit and serves as an excellent introduction to the process of nu-merical weather prediction. Platzman (1967) wrote that he found the illustrativeexample ‘one of the most interesting parts of the book’. In contrast to the straight-forward style of this chapter, much of the remainder of WPNP is heavy going,with the central ideas often obscured by extraneous material. It appears likely thatW. H. Dines advised Richardson to add the introductory example, in the hope thatit would make the key ideas of numerical prediction more accessible to readers.Richardson ends the chapter with an explicit acknowledgement: ‘I am indebtedto Mr W. H. Dines, F.R.S., for having read and criticised the manuscript of thischapter.’

4.1 Richardson’s model and data

The equations solved by Richardson were obtained by vertical integration of theprimitive equations linearized about an isothermal, motionless basic state. We

62

4.1 Richardson’s model and data 63

recall from (3.9) on page 48 that the vertical structure in this case is� �

�� , so that vertical integration is equivalent to multiplication by theequivalent depth � � �� . To facilitate comparison with Richardson’s treatment,we let � , � , � denote the vertically integrated momenta and pressure. The equa-tions (3.10)–(3.12) are formally unchanged by the integration:

� �� !

(4.1)� �� "�

� ��$# � !(4.2)

� ��$� ��%��'&)(+* � !-,(4.3)

The dependent variables are the vertically integrated horizontal momentum * �� /.0� � � � �/12�3 .4�/12�5 � and the vertically scaled pressure perturbation � � �7698where 6 8 is the surface pressure. The independent variables are the time

�, latitude:

and longitude ; (distances eastward and northward on the globe are given by< �>=@?�ACB :D< ; and< # �E= <�:

),=

is the Earth’s radius, % is the gravitationalacceleration and � �EFHGIBKJML :

is the Coriolis parameter. The scale height � �� ON@POQ� % was set at 9.2 km, corresponding to a mean temperature PRQ �FHFTS

K (Richardson denoted � by �VU and based this value on observations made byhis colleague, W. H. Dines). The reference density 12 may be determined by fixingthe mean surface pressure; taking 6 Q �XWY!H!H!

hPa it is approximately 1.1 kg m Z�[ .Richardson used the C.G.S.-system of units; the momentum values on page 8 ofWPNP are approximately equivalent to the SI values of wind speed multiplied by10 \ ; his pressure values are numerically equal to pressure in microbars or deci-Pascals.

The equations (4.1)–(4.3) are mathematically isomorphic to the linear shallowwater equations that govern the small amplitude dynamics of an incompressiblehomogeneous shallow fluid layer on a sphere. In this case one interprets � as themean depth and equates 6 8 to 12^] U where ] U � % �HU is the geopotential perturbationof the free surface. The solutions of the Laplace tidal equations were discussed,using both analytical and numerical methods, in the previous chapter.

The initial pressure perturbation chosen by Richardson is depicted in Fig. 4.1. Itis a simple zonal wavenumber one perturbation given by

6 8 �_WY!7`�BKJMLba : ?�ACB : BKJcL ; �edgf B4? fHh B � (4.4)

which is symmetric about the equator with maxima of magnitude 38.5 hPa ati !kjEast and lHl j North and South, and corresponding minima in the western

hemisphere at the antipodes of the maxima. The initial winds were taken to bein geostrophic balance with the pressure field and the total column momenta are

64 The barotropic forecast

Fig. 4.1. The initial pressure field chosen by Richardson for his barotropic forecast (repro-duced from WPNP, p. 6).

given (in kg m Z�� s Z�� ) by

� � WY! ` �FHG = � FR?�ACB a : BKJML a�: � BKJML ; . � � � WY! ` �FHG = BKJcL : ?�ACB ; , (4.5)

The maximum velocity is about 20 m s Z�� zonally along the equator. Fig. 4.1 isan ‘elevation’ view of the eastern hemisphere on what appears to be a globular orequidistant projection. It is reproduced from page 6 of Richardson’s book and wasalso featured on the cover of the Dover edition of WPNP.

4.2 The finite difference scheme

Richardson calculated the initial changes in pressure and wind (or momentum) us-ing a finite difference method. He selected a time step of 2700 seconds or [` h, andused a spatial grid analogous to a chess-board, with pressure and momentum eval-uated at the centres of the black and white squares respectively. The frontispiece ofWPNP, reproduced in Fig. 1.6 on page 21, depicts such a grid, showing the chess-board pattern of alternating black and white squares. In this illustration, Richardsondisplaced the grid by two degrees west from that used in his introductory example,a subterfuge designed so that the grid points would coincide more closely with theactual distribution of observing stations in the vicinity of the United Kingdom.

4.2 The finite difference scheme 65

A grid with the staggered discretization of points proposed by Richardson is to-day called an E-Grid, following the classification introduced by Arakawa (Arakawaand Lamb, 1977). Platzman suggested the name ‘Richardson Grid’ and we willadopt that name. The distance between adjacent squares of like colour is 400 kmin latitude, with 64 such squares around each parallel. This implies grid intervalsofF�� ; � l ,��HF l j ,

F�� : ��,��kj. The resolution at l !�j N is about the same in both

directions: the grid cells are approximately squares of side 200 km. A power of 2was selected by Richardson for the number of longitudes, to permit repeated halv-ings of resolution towards the poles, where the meridians converge. A sphericalgrid system having almost homogeneous grid density over the globe was later pro-posed by Kurihara (1965), who made reference to WPNP, and the current ECMWFmodel uses a reduced Gaussian grid, with a decreasing number of points aroundeach parallel of latitude as the poles are approached.

The advantage of the chess-board pattern is that the time-rates are given at thepoints where the variables are initially tabulated. This is illustrated by considera-tion of the discretization of the continuity equation.1 Pressure is evaluated at thecentres of the black squares and winds at the centres of the white. We write (4.3)in finite difference form as:� ��$� � %^�=/?�ACB :

� � �� ; �� ?�ACB : �� : � � !-,

(4.6)

Thus, to compute the tendency, we need the divergence on the black squares. Forthis, we require the zonal wind in the white squares to the east and west, andthe meridional wind in the white squares to the north and south of each blacksquare. These are provided by the gridded values. A selection of values is givenin Table 4.1. This is extracted from Richardson’s Table on page 8 of WPNP (weconvert his C.G.S.-units to SI units by division by ten). The spatial grid incrementsare

� ; � !^, ! i�� Fradians and

� : � !^, !�HF �radians. There are two terms in the

divergence: � �� ; � �� ; � F�,MW�� WY!7`� � � ?�ACB : �� : � �� ?�ACB : � �'8 ?�ACB : 8� : � W7, � l � � WY!7` ,We note that they are both negative; there is no tendency for cancellation betweenthem. The momentum divergence is given by

& ( * � W=/?�ACB : Q

� � �� ; �� ?�ACB :� : � � i , i W�� WY! Z�[

(= � F � WY!� �� ,�� WY!��

m) and the surface pressure tendency follows

1 The time discretization will be discussed in � 4.4 and in Chapter 5.


Table 4.1. Initial momentum values required to calculate the tendency at: � l !^, S�� , ; � !��. These values are extracted from the Table on Page 8 of WPNP

and converted to SI units.

� ��

�� "!#� $0.6129 P % �&�('*)��,+-��! ./�

P

� � ��,��! ./$0.6374 0 �213� 4 .�! . .

P 0 � �2 513� 4,.�! .-.

�768�9.�)�! 4 $0.6613 P % 68�(':.-;��-'"! � ;

P

immediately:� 6�8�$� � %�& ( *=< i ,?> i � � i , i W�� WY! Z�[ � � i ,?>T! � � WY! Z a Pa s Z�� ,

This corresponds to a change of pressure over the time step� � � [` h of 2.621 hPa,

in complete agreement with WPNP. This change may now be added to the initialpressure to obtain the pressure at the same position [` h later.

In a similar manner, considering the momentum equations (4.1) and (4.2), thegrid structure ensures that the values required to calculate the Coriolis term and thepressure gradient term are provided at precisely those points where the momentathemselves are specified — the white cells — so the updating yields values atthose points. Thus, the integration process can be continued without limit and is,in Richardson’s words, ‘lattice reproducing’. The chess-board pattern is depictedin Fig. 4.2 (after Platzman, 1967). The Richardson grid is composed of two C-grids, one comprising the quantities marked by asterisks, the other comprising theremaining quantities.

In replicating Richardson’s results, there was some difficulty in obtaining com-plete agreement. The discrepancy was finally traced to Richardson’s definition ofthe Earth’s rotation rate. He specifies the value

FHG � W7, S l � SkF�� WY! Z ` s Z�� onp. 13 of WPNP. This is correct. The lazy man’s value

FHG Q � W7, S l SHSHS$W � WY! Z ` s Z��is frequently used. It is defined using

G Q � F �� ^� FTS � �7! � �7! � , which implies a

4.3 The barotropic forecast 67

Fig. 4.2. The Richardson grid. Pressure is evaluated at the centres of the black squaresand winds at the centres of the white. The Richardson grid is composed of two C-grids,one comprising the quantities marked by asterisks, the other comprising the remainingquantities (Figure from Platzman, 1967).

rotation in a solar day. In fact, the Earth makes one revolution in a sidereal day,giving

Ga slightly higher value. It was characteristic of Richardson to use the more

accurate value.

4.3 The barotropic forecast

The changes in pressure and momentum were calculated by Richardson for a se-lection of points near England and entered in the Table on page 8 of WPNP. Thefocus here will be on the pressure tendency at the gridpoint 5600 km north of theequator and on the prime meridian ( l !^, S j N,

! jE). The value obtained by Richard-

son was a change of 2.621 hPa in [` hour. This rise suggests a westward movementof the pressure pattern (similar rises were found at adjacent points). Such a move-ment is confirmed by use of the geostrophic wind relation in the pressure tendencyequation (4.3). The momentum divergence corresponding to geostrophic flow is

& (Y*�� a� �� ,

(4.7)

This implies convergence for poleward flow and divergence for equatorward flow.Since the initial winds are geostropic, the initial tendency determined from (4.3) is� 6�8�� %�� a

� 6�8�� (4.8)


(WPNP, Ch. 2#8). On the basis of his calculated tendency, Richardson commentsthat ‘The deduction of this equation has been abundantly verified. It means thatwhere pressure increases towards the east, there the pressure is rising if the windbe momentarily geostrophic’ (WPNP, p. 9). Richardson argues that this deductionindicates that the geostrophic wind is inadequate for the computation of pressurechanges.

The right hand side of (4.8) is easily calculated for the chosen pressure perturba-tion (4.4). The parameter values are

= � F � WY! � �� m, % � i ,?> im s Z a , � � i , F

km,FHG �)W7, S l �� WY! Z ` s Z�� , in agreement with WPNP, p. 13. At the point in questionthe tendency calculated from (4.8) comes to

!^, ! i >�W��Pa s Z�� or 2.623 hPa in [` h.

This analytical value is very close to the numerical value (2.621 hPa) calculatedby Richardson, indicating that errors due to spatial discretization are small. Theassumption of geostrophy implies zero initial tendencies for the velocity compo-nents; this follows immediately from (4.1) and (4.2). The calculated initial changesof momentum obtained by Richardson were indeed very small, confirming the ac-curacy of his numerical technique.

Richardson, pointing out that ‘actual cyclones move eastward’, described a re-sult of observational analysis that showed a negative correlation between

� 6 8 ��and

� 6 8 �$ , in direct conflict with (4.8). In fact, the generally eastward move-ment of pressure disturbances in middle latitudes is linked to the prevailing strongwesterly flow in the upper troposphere. Presumably, Richardson was unaware ofthe dominant influence of the jet stream, whose importance became evident onlywith the development of commercial aviation. However, he certainly was aware ofthe prevaling westerly flow in middle latitudes, and with its increase in intensitywith height. Indeed, on page 7 of WPNP, he refers to Egnell’s Law, an empiricalrule stating that the horizontal momentum is substantially constant with height inthe tropopause. This implies an increase in wind speed by a factor of roughly threebetween low levels and the tropopause.

Richardson was free to choose independent initial fields of pressure and wind,but ‘it has been thought to be more interesting to sacrifice the arbitrariness in orderto test our familiar idea, the geostrophic wind, by assuming it initially and watch-ing the ensuing changes’ (WPNP, p. 5). Recall that he worked out the introductoryexample in 1919, after his baroclinic forecast, the failure of which he ascribed to er-roneous initial winds. He wanted to examine the efficacy of geostrophic winds and,moreover, equation (4.8) which follows from the geostrophic assumption allowedhim to compare the results of his numerical process with the analytical solutionfor the initial tendency. As has been seen above, the numerical errors were negli-gible. However, the implication of (4.8) for the westward movement of pressuredisturbances led him to infer that the geostrophic wind is inadequate for the calcu-lation of pressure changes. Towards the end of the chapter he concludes: ‘It has

4.3 The barotropic forecast 69

been made abundantly clear that a geostrophic wind behaving in accordance withthe linear equations . . . [(4.1)–(4.3)] cannot serve as an illustration of a cyclone’.He then questions whether the inadequacy resides in the equations or in the initialgeostrophic wind, or in both, and suggests that further analysis of weather maps isindicated.

Was Richardson’s conclusion on the inappropriateness of geostrophic initial con-ditions justified? It seems from a historical perspective that the reason that his sim-ple forecast lacked realism lies elsewhere. The zonal component of the geostrophicrelationship is obtained by omitting the time derivative in (4.2). Taking the verti-cal derivative of this, after division by �/12 , and using the hydrostatic equation andthe logarithmic derivative of the equation of state one obtains the thermal windequation:

� 3 �� %�'P � � P��# �� 3 �P � � P� � , (4.9)

The second right hand term may be neglected by comparison with the first. Takinga modest value of 30 K for the pole to equator surface temperature difference, andassuming that this is representative of the meridional temperature gradient in thetroposphere, (4.9) implies a westerly shear of about 10 m s Z�� between the surfaceand the tropopause. Thus, if the winds near the surface are light, there must bea marked westerly flow aloft. Although the importance of the jet stream was un-known in Richardson’s day, the above line of reasoning was obviously available tohim. Yet, he focussed on the geostrophic wind as the cause of lack of agreementbetween his forecast and the observed behaviour of cyclones in middle latitudes.The real reason for this was his assumption of a barotropic structure for his distur-bance, in contrast to the profoundly baroclinic nature of extratropical depressions.

Richardson might have incorporated the bulk effect of the zonal flow on hisperturbation by linearizing the equations about a westerly wind, 13 . This wouldresult in a modification of (4.8) to

� 6 8�� %^� �� a 13 � � 6 8�� ,

It would appear that, for sufficiently strong mean zonal flow, the sign of the pres-sure tendency would be reversed. However, things are not quite so simple. Theeffect of a background zonal flow is not a simple Doppler shift. The meridionaltemperature gradient that appears in the lower boundary condition tends to coun-teract the Doppler shift of the zonal flow. This ‘non-Doppler effect’ was discussedby Lindzen, 1968 (see also Lynch, 1979).


4.4 The global numerical model

Let us now consider the results of repeating and extending Richardson’s forecastusing a global barotropic numerical model based on the linear equations used byhim, the Laplace tidal equations. Precisely the same initial data were used, andthe same time step and spatial grid interval were chosen. There were only twosubstantive differences from Richardson’s method. First, an implicit time-steppingscheme was used to insure numerical stability, in contrast to the explicit schemeof WPNP which is restricted by the Courant-Friedrichs-Lewy criterion (see � 5.3below). Second, a C-grid was used for the horizontal discretization. The C-gridhas zonal wind represented half a grid step east, and meridional wind half a gridstep north of the pressure points. This grid was chosen when the model was firstdesigned, for reasons which are irrelevant in the current context. Richardson’s gridcomprises two C-grids superimposed, and coupled only through the Coriolis terms(Fig. 4.2). The initial pressure tendency calculated on one C-grid is independent ofthe other one.

The numerical scheme used to solve the shallow water equations is a linear ver-sion of the two-time-level scheme devised by McDonald (1986). The momentumequations (4.1) and (4.2) are integrated in two half-steps �a � � . In the first half-step,the Coriolis terms are implicit and the pressure terms explicit:

� �� a � � � � � �� .� � �� a � � �D� � �� ,

In the second half-step, the Coriolis terms are explicit and the pressure terms im-plicit:

� � �� a � � � � � �� .� �� a � � �D� � �� ,

The Coriolis terms are said to be pseudo-implicit, because the intermediate values�� and � �� can be eliminated immediately, yielding

� �� a � � � �� . (4.10)

� �� a � � � �� . (4.11)

where the right-hand terms � � � � � . �� depend only on the variables at time� � � . An expression for the divergence follows immediately:

& (Y* �� a � � & a � �� & (�� ,(4.12)

The continuity equation (4.3) is integrated in a single centered implicit step� �

:

� � �� a � � %�� & (+* �� "& (+* � � , (4.13)

4.5 Extending the forecast 71

Elimination of the divergence between (4.12) and (4.13) yields an equation for thepressure: �

& a �� a�� . (4.14)

where � a � S ^� � � a %�� and the right-hand forcing��

is known from variables attime � � � . This Helmholtz equation is solved by the method of Sweet (1977). Once� �� is known, � �� and � �� follow from (4.10) and (4.11). The simple formof (4.14) is a consequence of splitting the integration of the momentum equationsinto two half-steps. However, this splitting is formal: no � � � �a � � � values areactually computed. An analysis by McDonald (1986) showed the scheme to beunconditionally stable and to have time truncation � � � � a � .

4.5 Extending the forecast

The pressure and wind selected by Richardson and specified by (4.4) and (4.5)are shown in Fig. 4.3(a). The forecast fields of pressure and wind valid five dayslater may be seen in Fig. 4.3(b). Only the Northern Hemisphere is shown, as thefields south of the equator are a mirror image of those shown. The initial changein pressure at the location l !^, Sbj N,

!kjE was 2.601 hPa in [` hour. This is close

to Richardson’s value (2.621 hPa), the slight difference being due to the implicittime-stepping scheme. The wavenumber one perturbation rotates westward almostcircumnavigating the globe in five days. The pattern is substantially unchanged inextratropical regions. In the tropics a zonal pressure gradient, absent in the initialfield, develops and is maintained thereafter.

One may ask why the chosen perturbation is so robust, maintaining a coherentstructure for such a long time and undergoing little change save for the retrogressiveor westward rotation. This behaviour is attributable to the close resemblance ofRichardson’s chosen data to an eigensolution of the linear equations. The normal-mode solutions of the Laplace tidal equations were studied in the previous chapterand it was found that the solutions fall into two categories, the gravity-inertia wavesand the rotational modes. Haurwitz (1940) found that, if divergence is neglected,the frequencies of the rotational modes are given by the dispersion relation

� FHG�� W � (4.15)

where

and � are the zonal and total wavenumbers. For � W

and � � Fthis

gives a period of three days. The stream function of the RH(1,2) mode is� � � Q� �� a � ; . : �� Q BKJML : ?�ACB : BKJcL ; (4.16)

and we notice that this is related to Richardson’s choice of pressure through the


Fig. 4.3. (a) The initial pressure and wind fields in the Northern hemisphere as specifiedby Richardson. (b) Forecast pressure and wind fields valid five days later. (From Lynch,1992)

simplified linear balance relationship 6 8 � � 12 � . The wind �� & � is, of

course, nondivergent. But the geostrophic wind �� derived from 6 8 is not. The twoare related by

� �� 7� ��

where � is an eastward-pointing unit vector. Thus, the zonal wind chosen byRichardson differs from that of RH(1,2) and gives rise to non-vanishing diver-gence. The pressure and meridional wind components are identical: the structuresof the two waves are shown in Fig. 4.4. The period of RH(1,2) is three days but,as we saw in the previous chapter, the period of the corresponding Hough mode isapproximately five days.

The horizontal structure of the five-day wave is shown in Fig. 4.5(a) (with max-imum amplitude of 38.5 hPa as for Richardson’s field). The global model wasintegrated for five days using this wave as initial data and the result is shown inFig. 4.5(b). It is practically indistinguishable from the initial field except for theangular displacement (it has almost completed a full circuit of the globe), consis-tent with the expected behaviour of an eigenfunction. The initial pressure tendencycalculated for the five-day wave at the gridpoint at l !^, S j N,

! jE was 1.39 hPa in

[` h. Recall that Richardson’s data produced an initial change of 2.62 hPa. Whythe difference? Suppose that the pressure pattern (4.4) defined by Richardson wereto rotate westward without change of form, making one circuit of the Earth in a

4.5 Extending the forecast 73

−90 −60 −30 0 30 60 90−2

−1.5

−1

−0.5

0

0.5

1

Latitude

(a) Richardson’s Wave

−90 −60 −30 0 30 60 90−2

−1.5

−1

−0.5

0

0.5

1

Latitude

(b) Rossby−Haurwitz Wave

Fig. 4.4. (a) Richardson’s Initial Conditions. (b) First symmetric Rossby-Haurwitz waveRH(1,2). Dashed lines � , dotted lines � , solid lines

�. The pressures and meridional winds

are identical in (a) and (b); only the zonal winds differ.

period �� l days. It could then be described by the equation

6 8 � WY!7` B4JMLba : ?�ACB : BKJML � ; � F � � �� (4.17)

and the rate-of-change is obviously� 6 8�$� � � F �

�� WY! ` BKJML a : ?�ACB : ?�ACB � ; � F � �

�� (4.18)

which, at the point l !^, S j N,! j

E at time� � !

, gives the value 1.44 hPa in [` h,about half the value obtained by Richardson. One must conclude that the evolutionof his pressure field involves more than a simple westward rotation with period � .

An analysis of Richardson’s initial data into Hough mode components was car-ried out by Lynch (1992). It was shown that almost 85% of the energy projects ontothe five-day wave and about 11% onto the Kelvin wave. Over 99% of the energyis attributable to just three Hough mode components. Although relatively little en-ergy resides in the gravity wave components, they contribute to the tendency dueto their much higher frequencies. The temporal variations can be accounted for bythe primary component in each of the three categories: rotational, eastward gravityand westward gravity waves. The normal mode analysis of the fields used for thebarotropic forecast indicated the presence of a westward travelling gravity wavewith a period of about 13.5 hours. This component could be clearly seen in thetime-traces of the forecast fields (Lynch, 1992, Fig. 8). From a synoptic viewpoint,the short-period variations due to the gravity wave components may be considered


Fig. 4.5. The five-day wave, the gravest symmetric rotational eigenfunction (a) The initialpressure and wind fields in the Northern hemisphere. (b) Forecast pressure and wind fieldsvalid five days later. The wave has almost completed a circuit of the globe. (From Lynch,1992)

as undesirable noise. Such noise may be removed by modification of the initialdata, a process known as initialization.

A simple initialization technique, using a digital filter, was applied to the ini-tial fields, and was successful in annihilating the high frequency oscillations with-out significantly affecting the evolution of the longer period components (Lynch,1992). Although the barotropic model is linear, this initialization technique mayalso be used with a fully nonlinear model. The baroclinic forecast which formsthe centrepiece of Richardson’s book was spoiled by high frequency gravity wavecomponents in the initial data. These gave rise to spuriously large values of thepressure tendency. Since the design of the low-pass filter and its application tothe initialization of Richardson’s multi-level forecast will be described in a laterchapter, we will not discuss it further here.

4.6 Non-divergent and balanced initial conditions

Richardson recognized that the divergence field calculated from his data was un-realistic, and he blamed this for the failure of the baroclinic forecast. However,even if the wind fields were modified to remove divergence completely, high fre-quency gravity waves would still be present, and cause spurious oscillations. Wewill consider the baroclinic forecast in depth later. For the present, we investigatethe effectiveness of using nondivergent initial winds for the single-level forecast.Two 24-hour forecasts were carried out: one started from Richardson’s data as

4.7 Reflections on the single layer model 75

specified in (4.4) and (4.5); the other started from the same pressure field (4.4) butwith wind fields derived from the stream function (4.16). The pressure is related to�

by the linear balance relationship 6 8 � � 12 � . The latter conditions correspondto the Rossby-Haurwitz wave RH(1,2). The latitudinal structure of the two sets ofinitial conditions was shown in Fig. 4.4.

The pressure and meridional wind component are identical for the two initialstates; only the zonal winds differ. The level of high-frequency noise can be es-timated by calculating the mean absolute pressure tendency. This is equivalent,through the continuity equation, to the mean absolute divergence, which we denoteby � � . The evolution of � � for the two forecasts is shown in Fig. 4.6. The solidline is for the forecast from Richardson’s initial fields. The dashed line is for aforecast starting with the Rossby-Haurwitz structure. Obviously, the divergencefor the RH(1,2) conditions vanishes at the initial time. However, it does not remainzero: the RH wave is a solution of the non-divergent barotropic vorticity equation,but it is not an eigensolution of the Laplace tidal equations. We see that after aboutthree hours the divergence has grown from zero to a value comparable to that forRichardson’s data. Thus, setting the divergence to zero at the initial time is not aneffective means of ensuring that it remains small.

To demonstrate the beneficial effect of using properly balanced initial conditions,a third forecast was done, with the five-day wave specified as initial data. Theamplitude was scaled so that the maximum pressure was the same for all threeforecasts. The horizontal structure of these initial conditions was shown in Fig. 4.5.The five-day wave is a pure rotational mode, so that there is no projection onto thehigh-frequency gravity waves. The evolution of the noise parameter � � for thethird forecast is shown by the dotted line in Fig. 4.6. We see that it remains more-or-less constant throughout the 24 hour forecast, and at a level somewhat less thanthat of the other two forecasts (if numerical errors were completely absent, it wouldremain perfectly constant). This confirms that it is possible to choose initial datafor which the ensuing forecast is free from spurious noise, but that a non-vanishingdivergence field is required for balance. This issue will be examined in greaterdetail in Chapter 8, when normal mode initialization will be considered.

4.7 Reflections on the single layer model

With hindsight, we can understand Richardson’s introductory example as a remark-able simulation of the ‘five-day wave’. This wave, the gravest symmetric rotationalnormal mode of the atmosphere, has been unequivocally detected in atmosphericdata (Madden and Julian, 1972; Madden, 1979). It is almost always present, isglobal in extent with maximum amplitude in middle latltudes, extends at least tothe upper stratosphere and has no phase-shift with either latitude or height. Geisler


0 3 6 9 12 15 18 21 240

20

40

60

80

100

120

140

Time (hours)

108 ×

||∇

.V||

Mean Absolute Divergence

RichardsonRossby−HaurwitzFive−day Wave

Fig. 4.6. Mean absolute divergence ( �1 ��

s ��) for three 24-hour forecasts. Solid line:

Richardson’s initial conditions. Dashed line: Non-divergent initial conditions (Rossby-Haurwitz mode, �

� 1, �

� �). Dotted line: pure Hough mode initial data (five-day

wave).

and Dickinson (1976) modelled the five-day wave in the presence of realistic zonalwinds and found that its period was remarkably robust, being little affected by thebackground state because the zonal flow and temperature gradient at the Earth’ssurface produce almost equal but opposite changes in period. Rodgers (1976)found clear evidence for the five-day wave in the upper stratosphere.

Richardson regarded the westward movement of his pressure pattern to be inconflict with the generally observed eastward progression of synoptic systems inmiddle latitudes, and concluded that ‘ ... use of the geostrophic hypothesis wasfound to lead to pressure-changes having an unnatural sign’ (WPNP, p. 146). Healso neglected the dominant influence of the westerlies in the upper troposphere.Although the jet-stream was yet to be discovered, the existence of westerly shearfollows from simple balance considerations. Ironically, had Richardson includeda realistic zonal flow component in his introductory example, he would still havefound that the pressure disturbance moved westward, due to the robustness of itsperiod (Lynch, 1979).

Richardson was unaware of the importance of the five-day wave and other such

4.7 Reflections on the single layer model 77

free waves in the atmosphere. Although these solutions had been studied muchearlier by Margules and Hough, their geophysical significance became apparentonly with the work of Rossby (1939). Richardson made reference to Lamb’s Hy-drodynamics, where tidal theory was reviewed, but he considered (WPNP, p. 5)that ‘its interest has centered mainly in forced and free oscillations, whereas nowwe are concerned with unsteady circulations’. He was convinced that the singlelayer barotropic model was incapable of providing a useful representation of atmo-spheric dynamics.

Richardson and Munday (1926) carried out a thorough and meticulous investi-gation of data from sounding balloons to examine the ‘single layer’ question. Theyshowed, in particular, that if the pressure were everywhere a function of densityonly, then the atmosphere would behave as a single layer. The term barotropic hadbeen introduced by Vilhelm Bjerknes (1921) to describe such a hypothetical state.Richardson found nothing in the observations to justify the single layer assumptionand concluded that Laplace’s tidal equations were ‘a very bad description of ordi-nary disturbances of the European atmosphere’. The values of the dynamical height� estimated from the data were extremely variable and an assumption of constant� could not be justified. In the Revision File, inserted before page 23, there is acopy of a letter from Richardson to Harold Jeffreys, dated April 20, 1933, wherehe writes ‘The concept of a dynamical height of the atmosphere is a very greatidealization. . . . Any argument based on the fixity of � is rotten’.

It is unfortunate that Richardson argued so pursuasively against the utility of thesingle-layer model. As we will see, a simple barotropic model proved to be ofgreat practical value during the early years of operational numerical weather pre-diction. Richardson’s conclusions pertained only to the linear tidal equations. Theadvective process, embodied in the nonlinear terms, was not considered. More-over, the process of group velocity, whereby the energy of a disturbance can prop-agate eastward while individual wave components proceed westward, had yet tobe elucidated (Rossby, 1945). Richardson’s lack of confidence in the barotropicrepresentation of the atmopshere was shared by many of his colleagues, with theresult that operational numerical weather prediction in Britain was delayed untilbaroclinic effects could be successfully modelled. We will discuss this in moredetail in Chapter 10.

the barotropic forecast

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