baroclinic splitting of jetslydia/publications/2008-thompson-etal-… · a. thompson, l. stefanova,...

Meteorol Atmos Phys 100, 257–274 (2008)DOI 10.1007/s00703-008-0308-5Printed in The Netherlands

Department of Meteorology, Florida State University, Tallahassee, FL, USA

Baroclinic splitting of jets

A. Thompson, L. Stefanova, T. N. Krishnamurti

With 12 Figures

Received 14 May 2007; Accepted 18 July 2007Published online 18 August 2008 # Springer-Verlag 2008

Summary

Whether the split of the Somali jet, sometimes seen onmonthly mean streamline analyses, is a climatological or adynamical feature has been the subject of long-standingdebate. This paper explores the dynamical conditions leadingto a split jet within the framework of a simple barotropicdynamic system. The initial conditions for the dynamicalsystem, along with three other parameters – the jet width, thezonal wavelength, and the latitude of the �-plane, form aparameter space for the problem consisting of a range ofsolutions for the evolution of the jet. This paper identifiesa region in the parameter space in which these solutionssupport a splitting of the jet. The width and wavelengthof the Somali jet determined from observations are suchthat for most initial conditions the solutions reside near theboundary in parameter space between the split and non-split regions. It is therefore concluded that the splitting ofthe Somali jet can be a dynamical feature given the observedjet width and wavelength. Whether a split does or doesnot occur is determined by the parameters defining the initialzonal mean and perturbation flow in the jet, with the solutionbeing highly sensitive to these initial conditions.

1. Introduction

Figure 1 illustrates a typical climatology of Junewinds for the Arabian Sea at the 1 km level. Thisis based on Findlater (1971) where the surfaceand upper air conventional observations wereused. This illustration shows the strong wind

maxima offshore from Somalia. This low-leveljet is often labeled as the Somali jet. It has aclimatological wind intensity of approximately35 m=s and is located 1 km above the ocean. Inthis figure, note a downstream split of the jetinto two branches. One branch moves north to-wards central India and the other traverses southtowards Sri Lanka. This split has been an issuethat has drawn considerable interest. It has beendebated whether the split is entirely a climatol-ogical or dynamical feature Rao (1976). The cli-matological viewpoint is that over a certainnumber of days of the month the northern branchis active and over the other days of the month thesouthern branch is more intense. This picture canaffect the climatology by showing two branchesin the monthly mean. The other viewpoint is thatsometimes a dynamic instability of the Somalijet can occur leading to a real split of the jet nearSomalia creating two jets over India-Sri Lankalongitudes. A careful examination of daily chartsat the 850 hPa level, where a large number ofcloud track winds are available, supports the no-tion that two downstream jets can indeed coexiston a given day. Thus the concept of a dynami-cal split appears worthwhile to explore.

In the next illustration Fig. 2a, b two parcelsare shown from the 850 hPa level on July 16,1979 and July 1, 1979. One of these diagramsis a representation of a split jet and the other is a

Corrrespondence: Aarolyn Thompson, Department of Meteorol-

ogy, Florida State University, Tallahassee, FL 32306-4520, USA

(E-mail: [email protected])

normal, single jet. In Fig. 2b, the plotted low-levelcloud track winds clearly show two branches ofstronger intensity flows. These strong winds overthe east central Arabian Sea have an intensity onthe order of 50 knots. The other example Fig. 2aclearly shows a single wind maxima extendingfrom Somalia to the Indian coast. It should benoted, however, that there are many examplesof such analyses that prominently show thenorthern or the southern jet only on a given day.These latter examples also appear to correlatewith the wet and dry spells of the monsoon.When the northern jet alone is dominant, theIndian Monsoon tends to be wetter. When thesouthern jet alone is dominant the monsoon isdrier. The split jet seems to be a temporary fea-ture that lasts only a few days, but nevertheless ispresent and may be a feature of the transitionbetween a dry and a wet spell of the monsoon.

This paper is limited to exploring the dynam-ical conditions under which a split can or cannot

form within the framework of a simple barotro-pic dynamic (or horizontal shear flow dynamic)problem. For this purpose a simple lower orderspectral model is being used. In a simple dy-namical (nonlinear) system, Lorenz (1960), theconservation of vorticity allows for a three-com-ponent system on a beta plane. This well knownsystem carries three spectral amplitudes (A, F,and G). A coupled system of nonlinear equations(three equations and three unknowns) describesthe future evolution of these three spectral ampli-tudes. This low order dynamical system contains,as its basis function, a doubly periodic harmonicfunction. This system also carries the quadraticinvariance of total kinetic energy and enstrophy.This system of differential equations can easilybe integrated in time using a leapfrog time dif-ferencing scheme. This system illustrates theadvective effects on the future evolution of a sin-gle harmonic wave. An interesting modification ofthis simple system was proposed by Wiin-Nielsen

Fig. 1. Mean monthly airflowat 1 km in the month of Juneover Arabia. Thick linesemphasize the airstream. Thearrows show the directionof the airflow

258 A. Thompson et al.

(1961), which included a few more degrees offreedom. This system is capable of illustratingthe split of a zonal jet into two jets. Wiin-Nielsen (1961) formulated an initial state andan initial disturbance scale that allowed for sucha split. The purpose of this study is to examinethe possible barotropic splits and non-splits as aparameter sensitivity problem.

2. Statement of the problem

The goal of this study is to present the layout ofa simple dynamical system that allows the split-ting of a zonal jet. In this framework several

model parameters control the evolution of thejet. This study aims to illustrate the workingsof this framework and carry out a number ofsensitivity experiments for the parameter spaceof the problem. It is the goal of this study to showthat a region in a multidimensional space for arange of values of these parameters carries thesplit jets and that outside that region no split ofthe jet occurs in the simple dynamical model.

3. Numerical model

This study follows a dynamical model develop-ed by Wiin-Neilsen (1961). It is a modification

Fig. 2a. Example of a non-split jet over the Arabian Sea on July 1, 1979

Baroclinic splitting of jets 259

of the low order system of Lorenz (1960), whichincludes more degrees of freedom. Using the acro-nyms and symbols of Table 1, the basic equa-tions of this model are:

dB

dt¼ 2kQ½E1F3 � E3F1�

dC

dt¼ �2kQ½E1F3 � E3F1�

dE1

dt¼ k

�CF1 �

Q� 3

2ðQþ 1ÞBF1 �5Qþ 1

2ðQþ 1ÞBF3

� 7Q� 1

2ðQþ 1ÞCF3 �R

Qþ 1F1

�

dE3

dt¼ k

�ðBþ CÞF3 þ

3Q� 1

2ð9Qþ 1ÞBF1

� 15Q� 1

2ð9Qþ 1ÞCF1 �R

9Qþ 1F3

�

dF1

dt¼ �k

�CE1 �

Q� 3

2ðQþ 1ÞBE1 �5Qþ 1

2ðQþ 1ÞBE3

� 7Q� 1

2ðQþ 1ÞCE3 �R

Qþ 1E1

�

dF3

dt¼ �k

�ðBþ CÞE3 þ

3Q� 1

2ð9Qþ 1ÞBE1

� 15Q� 1

2ð9Qþ 1ÞCE1 �R

9Qþ 1E3

�; ð1Þ

Fig. 2b. Example of a split jet over the Arabian Sea on July 16, 1979. The split can be seen from the two areasof maximum wind speeds. These groups are also moving away from each other indicating a split


where

Q ¼ �2

k2

R ¼ �

k2

k is the zonal wave number and � is the meridio-nal wave number.

These six equations are for six unknowns B, C,E1, E3, F1 and F3. These are spectral (time depend-ent only) amplitudes for a barotropic model, i.e.,

@r2

@t¼ Jðr2 ; Þ � �

@

@x: ð2Þ

In the above equation an elementary function containing a zonal mean component � and aperturbation component 0 is used as the low-order stream function of the problem where

� ðy; tÞ ¼ DðBþ CÞ�

1 � y

D

�þ B

2�sinð2�yÞ

þ C

4�sinð4�yÞ; ð3Þ

0ðx;y; tÞ¼E1ðtÞk

sin�ysinkxþE3ðtÞk

sin3�ysinkx

þF1ðtÞk

sin�ycoskx

þF3ðtÞk

sin3�ycoskx: ð4Þ

These equations show how the unknown spec-tral amplitudes B, C, E1, E3, F1, and F3 enter the

problem. The integration domain for this prob-lem is double periodic. This is an initial valueproblem where it is essential to march in timewith the six variables of the system of Eqs. (1).This system allows one wave number k in thezonal direction and two wave numbers in themeridional direction, two because we wish tostudy the splitting possibility.

The solution of this problem is sensitive to thevalues of initial conditions for B, C, E1, E3, F1,and F3 and to the values of k and � that define theaspect ratio. A selection of values for these para-meters can dictate the split versus no split evolu-tion of a jet. The Wiin-Neilsen (1961) studyprovided an example of a split jet. The incentiveof the present study is to explore values of thesemodel parameters in order to map out a multi-dimensional parameter space while defining a pa-rameter space envelope of split and no split. Thewidth of the jet is represented by D where k ¼ �

Dand the wavelength of the jet is represented by Lwhere � ¼ 2�

L. The zonal mean wind and pertur-

bation wind are as follows:

Uðy; tÞ ¼ � @ �

@y¼ ðBþ CÞ � B cosð2�yÞ

� C cosð4�yÞ; ð5Þ

U0ðx; y; tÞ ¼ ��E1ðtÞk

cosð�yÞ sinðkxÞ

� 3�E3ðtÞk

cosð3�yÞ sinðkxÞ

� �F1ðtÞk

cosð�yÞ cosðkxÞ

� 3�F3ðtÞk

cosð3�yÞ cosðkxÞ: ð6Þ

It can be shown from the above formula that U issymmetric around the center of the jet channel�y ¼ D

2

�. When U has a single maximum in the

middle of the channel, the jet is not split. Whenthe maximum wind speeds are found concurrent-ly north and south of the center of the channel,the jet is considered split. Numerically, this isanalyzed by calculating the north=south deriva-tive of U:

@U

@y¼ 2�B sinð2�yÞ þ 4�C sinð4�yÞ: ð7Þ

@U@y ¼ 0 indicates the presence of a maximum orminimum. If two or more maxima are present asplit has occurred.

Table 1. A list of symbols and their definitions

Symbol Definition

Low-order stream function Zonal mean component of the stream function 0 Perturbation component of the stream functionU Zonal mean windU0 Perturbation component of the zonal windx Zonal coordinatey Meridional coordinateL Wavelength of the jetD Width of the jetB, C Spectral amplitudes defining the zonal meanE1, E3,F1, F3

Spectral amplitudes defining the perturbation

� Meridional wave numberk Zonal wave numberf Coriolis parameter� df=dyQ Ratio of �2 and k2

R Ratio of � and k2


4. Numerical experiments

The nine parameters of the low order spectralsystem are the initial conditions for B, C, E1,E3, F1, and F3, the jet width D, the wavelengthL, and �. If one were to consider a range of justten values for each parameter, the result wouldbe 109 different scenarios. Such a large amountof information would be almost impossible toanalyze, and just as impossible to present visual-ly. Therefore, instead of varying all nine pa-rameters simultaneously, consecutive experimentsare carried out in which five of the initial condi-tions and � are fixed at reasonable values, whilethe remaining parameters extend to a range oftwenty values each. These initial conditions wereselected from Winn Nielsen (1961). The resultsof these experiments are organized as maps ofsplit=no split regions as a function of D and Lfor different values of the varying initial condi-

tion amplitude. With that in mind the followingexperiments were carried out, see Table 2:

0 is the perturbation, is the zonal mean.

In this way the perturbation can affect themean and the mean can affect the perturbationbecause of the nonlinearity of the system ofEqs. (1). Change in the initial value of E1, E3,F1, and F3 will lead to change in the future valuesof B and C and vice-versa.

4.1 Experiment 1

Experiment 1 contains results (Figs. 3 and 4)where the initial values of B vary, and all otherparameters are held constant at t¼ 0. The nega-tive (positive) values of B represent the existenceof an easterly (westerly) jet. The westerly jet inFig. 1 (right-hand side) has a larger area of split-ting than that of an easterly jet (left-hand side).The split area appears to develop from the topleft corner of each graph, indicating that the com-bination of shorter wavelengths (L) and largerwidths of the jet (D) have more occurrences ofjet splits. Throughout the simulation the maxi-mum split occurrence is in the regions wherethe jet widths are between 2� 103–5� 103 kmand at smaller wavelengths. The area where thesplit occurs at negative values decreases gradual-ly while for positive values of B it increases grad-ually, excluding a notable transitional periodbetween values of �10 and 0 m=s. Figure 4 breaksdown the transitional window into segments of1 m=s and shows that as initial values of B in-crease, the instances of the jet split increase.Speaking only of B(0)>0 cases, for any givenvalue of B(0), the presence or absence of a splitis sensitive to the choice of D and L. For a smallL, a split is almost always present. Progressivelylarger D values allow for progressively larger Lvalues resulting in a split. This behavior con-tinues until D reaches a critical value, after whichlarger values of D allow for smaller L valuesagain resulting in a split. For jets with a widthof D�3–5� 103 km, depending on the value ofB(0), a split is present for the largest range of Lvalues. The ratio of D and L appears to be thedeciding factor for the existence or non-existenceof a split. For jets with a width of D� 6–6.5�103 km, the deciding factor is the jet wavelength.

Table 2. The experiments conducted with constants andvalues listed as well as the variables in the experiments

Experiment Variables Constants

1 B, L, D C¼�15 m=sE1¼ 25 m=sE3, F1, F3¼ 0 m=s�¼ 45�

2 C, L, D B¼ 30 m=sE1¼ 25 m=sE3, F1, F3¼ 0 m=s�¼ 45�

3 E1, L, D B¼ 30 m=sC¼�15 m=sE3, F1, F3¼ 0 m=s�¼ 45�

4 E3, L, D B¼ 30 m=sC¼�15 m=sE1¼ 25 m=sF1, F3¼ 0 m=s�¼ 45�

5 F1, L, D B¼ 30 m=sC¼�15 m=sE1¼ 25 m=sE3, F3¼ 0 m=s�¼ 45�

6 F3, L, D B¼ 30 m=sC¼�15 m=sE1¼ 25 m=sE3, F1¼ 0 m=s�¼ 45�


Fig. 3. Results from Experiment 1 where Bvaries while other spectral amplitudes areheld constant. Splits are indicated by areasof dark gray and non split regionsare in light gray


If it is shorter than 6� 103 km, there is a split andif it is longer than no split exists.

4.2 Experiment 2

Figure 5 through 7 contain results of Experi-ment 2, where all other amplitudes are held con-stant at t¼ 0 with the exception of amplitude C.For negative initial values of C there is a gradualdecrease in the overall split area with a largeincrease between 0 and 10 m=s. For all posi-

tive values of C the jet splits regardless of thevalues of L and D. Figure 6 focuses in on thetransitional window between the split and non-split regimes. A smooth decrease in split areaexists between initial C values of 0 and 6 m=s.However, between the values of 6 and 8 m=sthere is a transition from no split at any wave-length or width to split at every wavelength andwidth. Figure 7 explores the drastic transitionbetween 6 and 8 with the focus on C initialvalues between 7.1 and 7.6 m=s. The splits areoccurring at the shorter wavelengths and nar-

Fig. 4. Graphs showing the transitionfrom non-split to split regions from Fig. 3


Fig. 5. Results from Experiment 2 whereC varies while other spectral amplitudes areheld constant. Splits are indicated by areasof dark gray and non-split regionsare in light gray


rower jets while in the previous figure the splitsoccurred at shorter wavelengths and wider jets.Overall, these solutions are very sensitive to thevalues of C especially between 7 and 8 m=s. Asin Experiment 1 this experiment is also depen-dent on the values of D and L. Observing onlyC(0)�0 cases, when D is at a value of 3–4� 103 km, L can be any wavelength less than9� 103 km for certain values of C(0) and have asplit. However, as D becomes broader the valueof L becomes limited and splits occur less fre-quently. In the first figure of this experiment thelargest split region was at small wavelengths

and a width of approximately 3.5� 103 km. Inthe second figure of this experiment splits oc-curred most often at small wavelengths and widejets. Finally, in the third experiment the oppositeoccurs, the largest split region is located at smallwavelengths and narrow jets, thus emphasizingthe sensitivity of these solutions to the initialvalues of C.

4.3 Experiment 3

The graphs of this experiment display an interest-ing pattern. The graphs are symmetric around the

Fig. 6. Graphs showing the rapid transitionfrom non-split to split regions from Fig. 5


E1 value of zero. At zero the graph displays nosplit at any wavelength or width. This is due to thefact that if B and C are the only nonzero initialamplitudes then the following is concluded:

dB

dt¼ 0;

dC

dt¼ 0;

dE1

dt¼ 0;

dE3

dt¼ 0;

dF1

dt¼ 0;

dF3

dt¼ 0: ð8Þ

From the above equations it can be shown thatB¼B(t¼ 0), C¼C(t¼ 0), and E1¼E3¼F1¼F3¼ 0. Therefore if there is a split in the initialconditions it will persist throughout the experi-ment. It is also interesting to note that between

the values of �1 and 1 m=s the split region be-comes isolated. Excluding the 0.5� 103 km wave-length value the area becomes bounded at a widthof 4.5� 103 km and remains symmetrical aroundzero even at small numbers. Overall the graphsare consistent in that the largest split regionoccurs at smaller wavelengths and between D¼3–5� 103 km.

4.4 Experiment 4

This experiment shows an ongoing decrease ofthe split region for initial E3 values of �40 to�10 m=s, which changes and starts to increasebetween �10 and 0 m=s. From 0 to 40 m=s the

Fig. 7. Further observing the non-spit=split transition from Fig. 5


Fig. 8. Results from Experiment 3 whereE1 varies while other spectral amplitudesare held constant. Splits are indicatedby areas of dark gray and non-split regionsare in light gray


growth of the area stabilizes and only slightlyincreases. This experiment is slightly differentfrom the other experiments for it includes widejets in the split region whereas the other exper-

iments are consistent with a width of 2–5�103 km for the split region. Also, as in the pastexperiments, the jet splits typically occur withinthe region of smaller wavelengths.

Fig. 9. Graphs showing the transitionfrom non-split to split regions from Fig. 8


Fig. 10. Results from Experiment 4 whereE3 varies while other spectral amplitudesare held constant. Splits are indicatedby areas of dark gray and non-split regionsare in light gray

270 A. Thompson et al.: Baroclinic splitting of jets

Fig. 11. Results from Experiment 5 whereF1 varies while other spectral amplitudesare held constant. Splits are indicated byareas of dark gray and non-split regions arein yellow gray

Fig. 12. Results from Experiment 6 whereF3 varies while other spectral amplitudesare held constant. Splits are indicated byareas of dark gray and non-split regionsare in light gray

4.5 Experiment 5

The pattern of the split region in this experimentis very simple. For negative values there is adecrease in area and for positive values thereis an increase. Also, similar to Experiment 3the graphs are symmetrical around the initialvalue of zero, however in this case at an F1

initial value of zero there is a split region. Forthis experiment the splits occur most often atsmall wavelengths and widths of roughly 3–6� 103 km.

4.6 Experiment 6

The results from this experiment are quite similarto those from Experiment 5 with the exception ofone or two characteristics. The results of thisexperiment are not overly sensitive to the valuesof F3. There is a decrease in area between thevalues of �40 and 0 m=s and an increase in areabetween 0 and 40 m=s. As with the previous ex-periment there is also symmetry around the zerovalue. This experiment, however, has a greateroccurrence of splits at small wavelengths andwider jets up to 10� 103 km.

Discussion and conclusions

The numerical experiments that were conductedshow that the occurrence or non-occurrence of asplit in the jet is more sensitive to the initialvalues of B and C, which define the initial meanzonal flow, than to the initial values of E1, E3, F1,and F3, which define the initial perturbation. Inthis study, the split is defined in terms of themean zonal flow, so the above result is not sur-prising. The perturbation flow, however, alsoplays a role in the splitting of the jet, since thenon-linear equations contain terms representingthe interaction between the zonal mean and theperturbation amplitudes.

The split=no split solution is particularly sen-sitive to the sign of the initial value of B. Anegative B indicates an easterly jet, while a pos-itive B indicates a westerly jet. The numericalexperiments indicate that a split of the jet is farmore common for westerly jets, i.e., for positiveinitial values of B. Given the fixed set of para-meters, the solution is very sensitive to the ini-tial value of the amplitude C, particularly for

small positive values. Varying the initial valuesof C between 0 and 10 m=s, three different re-gimes are encountered – a regime in which theoccurrence of a split is decided by the values ofL and D, a regime in which there is no splitregardless of the values of L and D, and a regimein which a split occurs regardless of the valuesof L and D.

Of secondary importance is the sensitivity ofsolutions to the initial values of E1, E3, F1, andF3. When all perturbation amplitudes are zero,very small changes in E1 around zero can leadto a splitting of the jet, given appropriate valuesof L and D. The solution is also sensitive to thesign of the E3 initial amplitude. It has to be point-ed out that for all experiments excluding ex-periment 3, the initial value of E1 is non-zero,therefore in Experiments 4–6 the initial pertur-bation field is nonzero even if the varying ini-tial amplitude (E3 for Experiment 4, F1 forExperiment 5, and F3 for Experiment 6) is zero.This explains the lack of symmetry of the solu-tions in this case.

The solution is not very sensitive to the initialvalues of F1 and F3, although for larger values ofeither, a somewhat larger range of L and Dresults in a jet split. For both initial amplitudesthe solution is symmetric around 0 m=s, i.e., thesign of the initial amplitudes does not make adifference to the occurrence or non-occurrenceof a split. The numerical experiments show thatin most cases, given a set of initial amplitudes,the solution is sensitive to the values of the jetwavelength L and the jet width D. Most graphsof split=no split as a function of L and Dhave a characteristic shape. For widths smallerthan D� 2.5–4� 103 km, the split region isconfined to progressively longer wavelengthjets. For jet widths between D� 2.5–4� 103 kmand D� 5–6� 103 km, the split region is con-fined to progressively shorter wavelength jets.Finally, for jet widths greater than D� 5–6� 103 km there is a cut-off wavelength suchthat jets with wavelength shorter than it do split,and jets with wavelengths longer than it do notsplit.

How do these findings apply to the actualSomali jet? From Fig. 2a, it can be estimated thatthe width of the jet is about 20� of latitude andabout 50� of longitude. In other words, D� 2.2�103 km, and L� 5.5� 103 km. For the majority

A. Thompson et al.: Baroclinic splitting of jets 273

of experiments these values lie very close on ei-ther side of the boundary between the split andno-split regions, indicating a very strong sensitiv-ity of the split occurrence to the initial condi-tions. Thus it is concluded that the splitting ofthe Somali jet can indeed be a dynamical feature,whose manifestation depends on the exact param-eters defining the initial zonal mean and the ini-tial perturbation.

Acknowledgements

This research is funded by NSF grant No: ATM – 0241517.

References

Findlater J (1971) Mean monthly airflow at low levels overthe western Indian Ocean. Geophys Mem 115, H.M.S.O.,London, 53pp

Lorenz EN (1960) Maximum simplification of the dynamicequations. Tellus 12: 245–54

Rao YP (1971) Mean monthly airflow at levels over thewestern Indian Ocean. Geophys Mem 115, H.M.S.O.,London, 53pp

Rao YP (1976) Southwest Monsoon (Meteorological Mono-graph Synoptic Meteorology No. 1). Indian Meteorologi-cal Department, New Delhi, 367pp

Wiin-Nielsen A (1961) On short-and long-term variations inQuasi-Barotropic Flow. Mon Wea Rev 89: 461–476

274 A. Thompson et al.: Baroclinic splitting of jets

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