Balance regimes for the stability of a jet in an -plane shallow ......Fluid Dynamics Research 39...

Fluid Dynamics Research 39 (2007) 353–377

Balance regimes for the stability of a jet in an f-plane shallowwater system

Norihiko Sugimotoa,b,∗, Keiichi Ishiokaa, Shigeo YodenaaDepartment of Geophysics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto

606-8502, JapanbDepartment of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho,

Chikusa-ku, Nagoya, Aichi 464-8601, Japan

Received 16 May 2003; received in revised form 9 May 2006; accepted 25 July 2006

Communicated by Hiroshi Niino and Yoshiyuki Hayashi

Abstract

To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearlyin an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximationgives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in thenonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate ofunstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different betweenhigh and low Ro. In the low Ro regime (Ro < 5), geostrophic balance is dominant in the perturbation field, the ratio‖�‖/‖�‖ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr2/Ro, where � and � arevelocity potential and streamfunction, respectively. On the other hand, in the high Ro regime (Ro > 5), cyclostrophicbalance with basic shear is dominant, ‖�‖/‖�‖ ∝ Fr2. Considering that the barotropic instability is caused by theresonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio‖�‖/‖�‖ being small, different approximation of the linear shallow water equations for each regime is deduced.Properties of the linear unstable modes are explained with these approximations.© 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.

Keywords: Jet; Shear instability; Linear stability analysis; Nonlinear simulation; Shallow water system; Quasi-geostrophicapproximation; Balanced models

∗ Corresponding author. Department of Computational Science and Engineering, Graduate School of Engineering, NagoyaUniversity, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan.

E-mail address: [email protected] (N. Sugimoto).

0169-5983/$32.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.doi:10.1016/j.fluiddyn.2006.07.004

mailto:[email protected]

354 N. Sugimoto et al. / Fluid Dynamics Research 39 (2007) 353–377

1. Introduction

The aim of this study is to investigate the limit of validity of balanced models and the accuracy ofbalance relations in geophysical fluid dynamics. After Charney (1947) derived the quasi-geostrophicmodel, several balanced models such as semi-geostrophic model (Hoskins, 1975) and balance equations(Gent and McWillams, 1983a) have been proposed and used to investigate phenomena in geophysicalfluid systems. It is well known that these balanced models are valid only in some restricted parameterspace. However, the parameter space in which balanced models can be applied (or not) has not beenunderstood sufficiently. In the present study, as a first step to seek the parameter space in which balancedmodels are good approximations of a full geophysical fluid system, we use an f-plane shallow watersystem and its quasi-geostrophic approximation, which is one of the most important balanced models,and investigate the stability of a jet both linearly and nonlinearly for a wide range of parameter.

It is a fundamental problem in geophysical fluid dynamics that the limit of validity of balanced modelsis not clear. Observational studies have suggested that large-scale motions in the atmosphere and oceanscould be regarded as nearly balanced ones, such as geostrophic balance or gradient-wind balance. Usingthese balance relations, several balanced models which approximately describe balanced motion havebeen proposed (Gent and McWillams, 1983b). In these balanced models, fast gravity wave motions arefiltered out, and only prognostic parts of slow Rossby wave motions evolve with time. Because of thesimplicity of these models, our theoretical knowledge of slow motions depending on Rossby waves ispractically all acquired from these balanced models (Ford et al., 2000). However, since the limit of validityof balanced models has not been fully understood, misuse of these balanced models occurs sometimes.In addition, if we can confirm that a balanced model is a good approximation of the corresponding fullsystem even in the parameter range where conditions to deduce the balanced model are not satisfied, theusable range of the balanced model will be extended. Therefore, it is of great interest to know the limitof validity of balanced models.

In spite of the importance to know the limit of validity of balanced models, there is no comprehensiveview of the parameter space in which a balanced model is a good approximation of the full system.Traditionally, the accuracy of geostrophic or gradient-wind balance approximation has been studiedgenerally by scaling analysis of the primitive equation system (Gent and McWillams, 1983b), which is amore general fluid model. However, in formal scaling analyses, there is no answer about how the dynamicsis changed if the parameter needed for the balance exceeds the range of validity. To answer these questions,Spall and McWilliams (1992) studied numerical solutions of the shallow water system with random,balanced initial conditions for several parameters, including high Ro and high Fr. They found somebalanced flow regimes in the parameter space for their persistence of initial balance. Recently, Stegnerand Dritschel (2000) also investigated the stability of isolated shallow water vortices for a wide rangeof parameter. They showed a significant departure from quasi-geostrophic dynamics due to ageostrophicand large-scale effects. However, both study treated only a few parameter values. For this reason, thesestudies could not answer completely the parameter range in which a particular balanced model is a goodapproximation of the full system. It is an open question what happens if the parameter values are out ofthe range where the geostrophic balance holds in a formal scaling analysis.

Recently, there has been another important topic in the limitation of validity of balanced models, whichis related to gravity wave radiation from nearly balanced flow. Although all balanced models assumeno gravity wave radiation from slow Rossby modes, there are several papers which suggest gravitywave radiation from balanced motion (Ford, 1994; Vanneste and Yavneh, 2004). Recently, we performed

N. Sugimoto et al. / Fluid Dynamics Research 39 (2007) 353–377 355

nonlinear numerical simulation in forced-dissipative f-plane shallow water system to investigate gravitywave radiation from a balanced jet flow (Sugimoto et al., 2006). We showed that while gravity waves wereradiated continuously from unsteady motion of the balanced flow, amplitudes of gravity waves were verysmall. This result suggested that a balance could be maintained approximately in the nonlinear phase ofthe instability.

Therefore, in the present study, we will check the validity of a particular balanced model, quasi-geostrophic approximation, for the shallow water system which we employed in our previous study. Forthis purpose, we investigate the stability of a balanced jet flow both linearly and nonlinearly for a wideparameter range and a large number of parameter values of Ro and Fr. Linear stability in shallow watersystems has been studied by many authors, theoretically (Satomura, 1981; Ripa, 1983; Balmforth, 1999)or in geophysical interests (Orlanski, 1968; Kubokawa, 1985). However, the main motivation of thesepapers were to understand the characteristics of the instability modes or the stability properties appliedfor geophysical phenomena, so that these studies were restricted in only narrow parameter range andsmall number of parameter values. Therefore, we investigate the linear stability of the jet for a wideparameter range to check the validity of quasi-geostrophic approximation. Since our interests are notonly in the initial instability process but also in what happens in the nonlinear phase of instability, weconduct nonlinear numerical experiments for these unstable jets in both the shallow water system andits quasi-geostrophic approximation. Again, we study the limitation of validity of the balanced model(quasi-geostrophic approximation) for a wide parameter range in the nonlinear phase.

This paper is organized as follows. In Section 2, the results of linear stability analysis of a zonal jetare shown. The stability characteristics in a shallow water system and those in its quasi-geostrophicapproximation are compared for a wide range of parameters, and it is shown that the unstable modes inthe shallow water system have some balanced regimes. Nonlinear simulations are also performed in theshallow water system and its quasi-geostrophic approximation in Section 3, where we investigate whetherthe balance found in the linear stability analysis is maintained or not in the nonlinear phase. Discussionson the unstable modes in terms of resonance of Rossby waves are included in Section 4, and we showthe dominant balance in each regime by the use of linearized shallow water equations. Conclusions aregiven in Section 5.

2. Linear stability analysis

2.1. Model description

The basic equation used in this study is a shallow water equation in a rotating frame, which is thesimplest equation including both vortical motions and gravity wave motions. On a plane rotating with theangular velocity �, which is generally called f-plane, the equation takes the following nondimensionalform

�u

�t+ Ro

(u

�u

�x+ v �u

�y

)− v = − ��

�x, (1)

�v

�t+ Ro

(u

�v

�x+ v �v

�y

)+ u = − ��

�y, (2)


��

�t+ Ro

(u

��

�x+ v ��

�y

)+(

Ro2

Fr2+ Ro�

)(�u

�x+ �v

�y

)= 0, (3)

where dependent variables u and v are the longitudinal (x) and latitudinal (y) velocities, respectively, and �is the free surface deviation from the mean geopotential height �0 of the fluid. Nondimensional parametersare Rossby number Ro ≡ U/f L and Froude number Fr ≡ U/√�0, where U and L are the characteristicspeed and the length scale, respectively. External parameter f = 2� is the Coriolis parameter. Here wefix the characteristic time scale as T = 1/f . From the shallow water system (1)–(3) with three dependentvariables, the well-known quasi-geostrophic vorticity equation with a single dependent variable is derivedfor almost geostrophic flow (Ro>1) as

(�

�t+ ��g

�x

�

�y− ��g

�y

�

�x

)(∇2�g −

Fr2

Ro2�g

)= 0, (4)

where �g stands for the surface deviation in a geostrophic balance, and ∇2 is the horizontal Laplacian. Inthis system, gravity wave motions are removed, and if the solution of prognostic part �g in (4) is given,the flow of diagnostic part (ug, vg) is obtained from the geostrophic balance condition: ug = −��g/�yand vg = ��g/�x. From now on, we will call this approximated system of (4) as the quasi-geostrophicsystem.

The domain for numerical investigations is assumed to be infinite in longitudinal direction, and periodicin latitudinal direction. The basic state of a zonal jet which may satisfy the barotropic instability conditionis taken from Hartmann (1983) as

u∗(y∗) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

+U0 sech{

2(y∗ − y∗+)B

}− � (0�y∗��),

−U0 sech{

2(y∗ − y∗−)B

}+ � (��y∗�2�),

(5)

where U0, B, and y± are parameters to determine the intensity, the width, and the position of the jet,respectively. Here ∗ denotes dimensional variables. We assume two jets flowing in opposite directions tosatisfy the periodic boundary condition in latitudinal direction. In order to connect two opposite jets, weshift the amplitude of the jets slightly by the � terms to have zero value at the tails. The surface deviationprofile in the geostrophic balance with the zonal jet is given by

�∗(y∗) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−fBU0 arctan{

exp

(2(y∗ − y∗+)

B

)}(0�y∗��),

−fBU0 arctan{

exp

(2(y∗− − y∗)

B

)}(��y∗�2�).

(6)

The width B and the intensity U0 are used as the length and the speed scales, respectively, then thenondimensional parameters Ro and Fr are determined as

Ro ≡ U0f B

and Fr ≡ U0√�0

. (7)


Fig. 1. Latitudinal profiles of the basic state with no wavy structure in x direction (Ro = 1 and Fr = 0.33): (left) the zonal flowu(y), (center) the free surface displacement �(y), and (right) the latitudinal gradient of the potential vorticity dq(y)/dy in ydirection.

Linear stability characteristics of the jet are investigated for a wide range of these parameters. In thisstudy, f and B are fixed. Therefore, Ro is proportional to U0 and Fr is proportional to 1/

√�0 for the

same Ro.Fig. 1 shows an example of latitudinal profiles of the basic state for Ro = 1 and Fr = 0.33: the

nondimensional zonal flow u(y), free surface deviation �(y), and latitudinal gradient dq(y)/dy of thepotential vorticity (q = ( + f )/h; here = �v/�x − �u/�y is the vorticity) in y direction. This set ofRo = 1 and Fr = 0.33 is an interesting value, since gravity waves are radiated from a balanced jet flowin the nonlinear phase of instability (Sugimoto et al., 2006). Latitudinal domain is taken to be 20 timesof B, and the centers of the two jets flowing in opposite directions are set at the positions of y+ = 5 andy− = 15 in order to satisfy the periodic boundary condition in latitudinal direction.

Note that u is symmetric with respect to the center of the jet while � is asymmetric. That is, the to-tal depth of the fluid is shallowest at y = 10 and deepest at y = 0 and 20. Since dq(y)/dy changesits sign around the jet, this basic flow satisfies a necessary condition for the barotropic instability(Ripa, 1983).

2.2. Linearization and eigenvalue analysis

Let the dependent variables (u, v, �) be divided into the basic state (u(y), 0, �(y)) and small perturba-tions (u′, v′, �′). Substituting these variables into (1)–(3), and neglecting second-order terms with respectto the perturbations, we can obtain the linearized shallow water equations as

�u′

�t+ Rou�u

′

�x+ Rodu

dyv′ − v′ = −��

′

�x, (8)

�v′

�t+ Rou�v

′

�x+ u′ = −��

′

�y, (9)

��′

�t+ Rou��

′

�x+ Ro d�

dyv′ = −

(Ro2

Fr2+ Ro�

)(�u′

�x+ �v

′

�y

). (10)


Defining streamfunction �′ and velocity potential �′ as u′ =−��′/�y +��′/�x and v′ =��′/�x +��′/�y,we rewrite the linearized shallow water equations (8)–(10) for the dependent variables (�′, �′, �′) as

�∇2�′�t

+ Rou�∇2�′

�x− Rodu

dy∇2�′ − Rod

2u

dy2

(��′

�x+ ��

′

�y

)+ ∇2�′ = 0, (11)

�∇2�′�t

+ Rou�∇2�′

�x− 2Rodu

dy

�

�x

(��′

�x+ ��

′

�y

)− ∇2�′ + ∇2�′ = 0, (12)

��′

�t+ Rou��

′

�x+ Ro d�

dy

(��′

�x+ ��

′

�y

)+(

Ro2

Fr2+ Ro�

)∇2�′ = 0. (13)

Assuming a plane wave solution in the form of

{�′, �′, �′} = Re[{A�(y), A�(y), A�(y)}eik(x−ct)], (14)we can obtain eigenvalues c = cr + ici for each zonal wavenumber k. Thus, we can determine the growthrate kci and the phase velocity cr for each perturbation. In this study, we calculate eigenvalues andeigenfunctions numerically by a matrix method (QR method) after discretizing (11)–(13) into 256 layersin y direction. Similar eigenvalue analysis is also done for the linearized equation of the quasi-geostrophicvorticity equation (4),(

�

�t+ u �

�x

)(∇2�′g −

Fr2

Ro2�′g)

+(

Fr2

Ro2u − d

2u

dy2

)��′g�x

= 0. (15)

2.3. Results

First, we present the results of the typical parameter set of Ro = 1 and Fr = 0.33. Fig. 2 shows thegrowth rate and the phase velocity against the perturbation wavenumber in the shallow water system (left)and those in the quasi-geostrophic system (right). Both systems have two unstable modes; one has themaximum growth rate at k=0.2 (the first mode denoted by ◦), and the other has the maximum growth rateat k = 0.1 (the second mode denoted by +). The difference in the growth rate between the two systemsis less than 5%. The phase velocities of the unstable modes are also very close between the two systems.Thus the quasi-geostrophic system gives a good estimation for the characteristics of unstable modes forthis choice of the parameters. Neutral inertial gravity waves with fast phase velocities can be seen in theshallow water system, but these waves are irrelevant to the unstable modes. Since numerical calculationis performed in finite wavenumbers, there is a discontinuous behavior for the maximum growth rate atabout wavenumber k ∼ 0.4.

The eigenfunction of the most unstable mode (k = 0.2) for Ro = 1 and Fr = 0.33 is shown in Fig. 3, forthe shallow water system (top) and the quasi-geostrophic system (bottom). The streamfunction �′ in theshallow water system is quite similar to that in the quasi-geostrophic system. The spatial pattern is almostsymmetric with respect to the center of the jet in the shallow water system, while it is perfectly symmetricin the quasi-geostrophic system (see Appendix A for the detailed calculation of the eigenfunction ofthe quasi-geostrophic system). On the other hand, �′ and �′ have north–south asymmetry with respectto the center of the jet. They have larger amplitude on the north side where the fluid is shallower than


Fig. 2. Growth rate (top) and phase velocity (bottom) plotted against wavenumber for Ro = 1 and Fr = 0.33, in the shallow watersystem (left) and in the quasi-geostrophic system (right). The first unstable mode is indicated by “◦”, the second unstable modeby “+”, and neutral modes by “·”.

the south side as shown in Fig. 1 (center). The asymmetry of the basic surface deviation � brings aboutthe asymmetry in the perturbation fields in the shallow water system. In contrast, the basic state in thequasi-geostrophic system is completely symmetric, so that the perturbation field also has the symmetry.The amplitude of �′ is more than one order of magnitude larger than that of �′. That is, the rotationalperturbation �′ is dominant in this choice of the parameters. The phases of �′ and �′ are nearly thesame, while �′ is about 14 wavelength out of phase to �

′ and �′. This phase shift will be discussedin Section 4.

Since our interests are not only in particular geophysical phenomena, but in the parameter space inwhich the quasi-geostrophic system is valid, we investigate the linear stability and the structure of unstablemodes for a wide range of parameter space. Fig. 4 shows the growth rate and the phase velocity againstthe perturbation wavenumber for relatively high Ro (Ro = 100, Fr = 0.01, left) and low Ro (Ro = 0.1,Fr = 0.01, right) in the shallow water system. There are two unstable modes corresponding to Fig. 2 inboth parameters; one has the maximum growth rate at k = 0.2 , and the other has the maximum growthrate at k = 0.1. The wavenumber of the maximum growth rate is nearly the same in both parameters, andthe value of maximum growth rate is roughly proportional to Ro. Note that in the formal scaling analysis,relatively high Ro does not assure the validity of quasi-geostrophic system.

Next, we show the eigenfunctions of the most unstable mode for high Ro (=100; top) and low Ro(=0.1; bottom) for Fr = 0.01 in the shallow water system in Fig. 5. The streamfunction �′ has almost thesame structure as in Fig. 3 for both cases. However, �′ and �′ are not similar to those in Fig. 3. For highRo, �′ and �′ are roughly anti-symmetric with respect to the center of the jet, and the pattern of �′ is quite


Fig. 3. Eigenfunction of the most unstable mode (k = 0.2) for Ro = 1 and Fr = 0.33. One half of the computing region is shownin y direction. The jet flows eastward with its center at y = 5. Top: the shallow water system, �′ (left: contour interval 5 × 10−2),�′ (center: contour interval 1.2 × 10−3), �′ (right: contour interval 5 × 10−4). Bottom: the quasi-geostrophic system, �′, �′, and�′ (contour intervals are the same as in the shallow water system). Solid contour lines correspond to positive values and zero,while broken contour lines correspond to negative values.

different from that of �′. On the other hand, for low Ro, �′ and �′ are roughly symmetric, and �′ ∼ �′,which correspond to the pattern of the geostrophic balance. In both choices of the parameter values, theamplitude of �′ is more than one order of magnitude larger than that of �′.

To study the parameter range in which the quasi-geostrophic approximation is valid, the maximumgrowth rate of perturbations and its zonal wavenumber are investigated in both systems for a wide range ofRo (10−3 �Ro�103) and Fr (10−3 �Fr�10). Fig. 6 shows the maximum growth rate in Fr–Ro parameterspace. There are two areas which are beyond the scope of this paper. The black shaded area (Fr2/Ro�1)is nonexistence area of the basic state where �∗ is comparable or greater than �0. The gray shaded area(Fr2/Ro2 �1) is almost stable area where the basic flow is very weak. Since it is rather difficult to calculateeigenvalue correctly in this almost stable area, the present study does not cover this area. Similarly, becauseof the finite discretization in numerical calculation, we cannot obtain eigenvalue accurately in the shallowwater system at the marginal area where the depth of the fluid is very shallow.

The quasi-geostrophic system gives a good estimation for the maximum growth rate for Fr�1 andFr2/Ro2 �1, even for high Ro. In this regime, the maximum growth rate is almost proportional to Roand independent of Fr, since the growth rate for the barotropic instability is scaled by the shear of thebasic flow. High Ro corresponds to stronger jets, thus the maximum growth rate also becomes larger. Thewavenumber of the maximum growth is k = 0.2 in almost all the parameter ranges in both systems (notshown here).


Fig. 4. Growth rate (top) and phase velocity (bottom) plotted against wavenumber for Ro = 100, Fr = 0.01 (left) and Ro = 0.1,Fr = 0.01 (right) in the shallow water system. The meanings of the marks (◦, +, ·) are the same as Fig. 2.

When Fr > 1, the phase velocities of rotational modes become close to those of gravity wave modes.Then, the instability due to the resonance of Rossby waves and gravity waves can occur where the phasevelocities of Rossby waves and gravity waves coincide (Fig. 7). Fig. 8 shows an example of these unstablemodes. The disturbance of �′ is localized in the jet region, which is related to a Rossby wave. On the otherhand, in �′ and �′ fields, the disturbance has large amplitudes out of the jet region. The phase relationshipbetween �′ and �′ shows that the disturbance has the character of an eastward propagating gravity wave.Note that these unstable modes are not present in the quasi-geostrophic system.

Fig. 9 shows the relative deviation (max,sh −max,qg)/max,sh of the maximum growth rate in the quasi-geostrophic system from that in the shallow water system, where max,sh and max,qg mean the maximumgrowth rates of the shallow water system and those of the quasi-geostropic system, respectively. Thereare two different areas in the parameter space in which signs and parameter dependences of these valuesare different. In the area where RoFr�1 and Fr2/Ro2 �1, the relative deviation has a positive value anddepends on Fr/Ro; the deviation becomes smaller by one order when Ro becomes larger by one order,while it becomes smaller by one order when Fr becomes smaller by one order. On the other hand, in thearea where RoFr�1 and Fr�1, the relative deviation has a negative value and depends on Fr; the deviationis independent of Ro, and becomes smaller by one order when Fr becomes smaller by one order. Sincethe deviation changes its sign around the line RoFr = 1, this value has a local minimum along this line.

Since the deviation in Fig. 9 must be caused by the divergence component, we investigate the ratio of themaximum amplitude of the divergent flow ‖�′‖ to that of the rotational flow ‖�′‖ in Fig. 10. Here ‖·‖ meansL∞-norm. Over all parameter ranges in which the quasi-geostrophic approximation for the maximumgrowth rate is valid ((max,sh−max,qg)/max,sh �10%; Fr�1), the amplitude of ‖�′‖ is much smaller than


Fig. 5. Eigenfunction of the most unstable mode (k =0.2) for Ro=100 and Fr=0.01 (top), and Ro=0.1 and Fr=0.01 (bottom)in the shallow water system. �′ (left: contour interval 5 × 10−2), �′ (center: contour interval 1.5 × 10−7 for the top, 1 × 10−5for the bottom), and �′ (right: contour interval 1.5 × 10−2 for the top, 5 × 10−2 for the bottom). The meanings of solid andbroken contour lines are the same as Fig. 3.

Fig. 6. Diagram of the maximum growth rate of the most unstable mode in Fr–Ro parameter space. The shallow water system(left) and the quasi-geostrophic system (right). The marks (◦, �, �) indicate the parameters, ◦ (Ro = 0.1, Fr = 0.01), �(Ro = 100, Fr = 0.01), � (Ro = 100, Fr = 5), respectively.


Fig. 7. Growth rate (top) and phase velocity (bottom) plotted against wavenumber for Ro = 100, Fr = 5 in the shallow watersystem. The unstable modes are indicated by “◦” and neutral modes by “·”.

Fig. 8. Eigenfunction of the unstable mode (k = 0.2) for Ro = 100 and Fr = 5. �′ (left: contour interval 6 × 10−2), �′ (center:contour interval 2 × 10−2), and �′ (right: contour interval 6 × 10−3). The meanings of solid and broken contour lines are thesame as Fig. 3.

that of ‖�′‖, less than 5%. That is, the parameter range in which the rotational component �′ is dominantcorresponds to the range in which quasi-geostrophic system gives a good estimation for the maximumgrowth rate in the shallow water system. However, based on the contours of the ratio, we can divide thisregime into two sub-regimes. In the low Ro regime (Regime 1; Ro < 5), the ratio is proportional to Fr2/Ro;the ratio becomes larger by one order when Ro becomes smaller by one order, while it becomes larger bytwo orders when Fr becomes larger by one order. On the other hand, in high Ro regime (Regime 2; Ro > 5),


Fig. 9. Diagram of the relative deviation of the maximum growth rate in the quasi-geostrophic system from that in the shallowwater system ((max,sh − max,qg)/max,sh) in Fr–Ro parameter space. The meanings of the marks (◦, �, �) are the same asFig. 6. Solid contour lines correspond to positive values and zero, while broken contour lines correspond to negative values.

Fig. 10. Diagram of the ratio ‖�′‖/‖�′‖ of the most unstable modes in Fr–Ro parameter space. The meanings of the marks (◦,�, �) are the same as Fig. 6.


the ratio is proportional to Fr2; the ratio is independent of Ro, and becomes larger by two orders when Frbecomes larger by one order. This difference between these regimes must be explained, since this resultmay cause the parameter dependence of the relative deviation shown in Fig. 9. We will discuss about thisdifference in Section 4.

3. Nonlinear time-evolution

In the linear stability analysis in Section 2, we obtained a wide parameter range in which the quasi-geostrophic system gives a good estimation for the maximum growth rate of the most unstable modes in theshallow water system. In this section, we will check the validity of the quasi-geostrophic approximationin the nonlinear phase of the instability.

3.1. Model description

First, we show the model description of nonlinear time-evolution in this subsection. The shallow waterequations (1)–(3) are rewritten for the dependent variables (, �, �), which are the vorticity, the divergence(� = �u/�x + �v/�y) and the geopotential height (� = � + �0), respectively, as

�

�t= −�(u)

�x− �(v)

�y− f �, (16)

��

�t= �(v)

�x− �(u)

�y+ f − ∇2(E + �), (17)

��

�t= −�(u�)

�x− �(v�)

�y, (18)

where E = (u2 + v2)/2, and (u, v) are obtained from the streamfunction � and the velocity potential �defined by = ∇2� and � = ∇2�, that is, (u, v) = (−��/�y + ��/�x, ��/�x + ��/�y).

In the linear stability analysis in Section 2, the distance between the two jets is so long that the unstablemode around each jet is not affected by the presence of the other jet, which we have checked by changingthe distance. In the nonlinear phase of the instability, however, the unstable modes which have finiteamplitudes may interact unless the jets are separated enough. Therefore, in the numerical experiment inthis section, the length of the domain in the y direction is extended to be 8� which is 4 times longer thanthat in the x direction, 2�. By this setup, the two jets are separated so that they may not interact with eachother even in the nonlinear phase. The basic state of the zonal jet (5) and (6) are rewritten as

u∗(y∗) =

⎧⎪⎪⎨⎪⎪⎩

+U0sech{

2(y∗ − 2�)B

}− � (0�y∗�4�),

−U0sech{

2(y∗ − 6�)B

}+ � (4��y∗�8�),

(19)

�∗(y∗) =

⎧⎪⎪⎨⎪⎪⎩

−fBU0 arctan{

exp

(2(y∗ − 2�)

B

)}(0�y∗�4�),

−fBU0 arctan{

exp

(2(6� − y∗)

B

)}(4��y∗�8�).

(20)


A very small initial perturbation is added to the vorticity field as

I =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−AI exp{

−(x∗ − �)2 + (y∗ − 2�)2

2B2

}(0�y∗�4�),

AI exp

{−(x

∗ − �)2 + (y∗ − 6�)22B2

}(4��y∗�8�)

(21)

with AI = Ro × 10−5.The basic equations (16)–(18) are integrated numerically with a spectral transform code (Ishioka,

2002). The dependent variables are expanded as

W(x, y, t) =K∑

k=−K

L∑l=−L

skl(t)eikxeily/4, (22)

where W(x, y, t) represents , �, and �. K and L are truncation wavenumbers in x and y directions,respectively. If the time-evolution of , �, � in physical space is written as

�W

�t= Z(x, y, t), (23)

where Z represents the right-hand side of (16)–(18), then the time-evolution of the coefficient skl isdetermined by the forward transform as

dskldt

= 116�2

∫ 8�0

∫ 2�0

Z(x, y, t)e−ikxe−ily/4 dx dy. (24)

We set K = 84 and L = 336 with 256 × 1024 grids so that the grid intervals in x and y directions are thesame. The fourth-order Runge–Kutta method is used for time integrations with an increment of 0.005/Ro.We also introduce an artificial viscosity term �(∇2)5W for each dependent variable to smooth numericalbehavior, and set the viscosity coefficient � = Ro × 10−21. Similar nonlinear calculation with a spectralmethod is also done for the quasi-geostrophic vorticity equation (4).

3.2. Results

Fig. 11 shows an example of the time-evolution for Ro = 0.1 and Fr = 0.01 (Regime 1) in both thequasi-geostrophic system (left) and the shallow water system (right 3 figures). As it can be seen, thequasi-geostrophic system is still good approximation of the shallow water system in the highly nonlinearphase (t = 1100, 1520). Geostrophic balance is dominant between and � in the shallow water system.

Fig. 12 shows an example of the time-evolution for Ro = 100 and Fr = 0.01 (Regime 2) in boththe quasi-geostrophic system (left) and the shallow water system (right 3 figures). The quasi-geostrophicapproximation is still valid even in the highly nonlinear phase (t =1.10, 1.52) for high Ro. This is becausewhen the width of the shear layer is much smaller than the radius of deformation, � easily adjusts to thevelocity field at an expense of very tiny portion of the kinetic energy. This result clearly shows that thetime-evolution of the whole flow field is determined by the rotational flow . That is, � and � are merelyslave to the rotational flow . In this regime, the cyclostrophic balance is dominant between and � inthe shallow water system. Thus, strong vortex region corresponds to shallower depth region.


Fig. 11. Nonlinear evolution for Ro = 0.1, Fr = 0.01 in the quasi-geostrophic system (a) and the shallow water system (b; , c;�, d; �, respectively) at t = 1100 (top) and t = 1520 (bottom) in x–y field. Negative � is painted in pink and the contour intervalis 3 × 10−6 in (c).

Fig. 13 shows an example of the time-evolution for Ro = 100 and Fr = 0.5 (Regime 2) in both thequasi-geostrophic system (left) and the shallow water system (right 3 figures), which is for relatively highFr. The quasi-geostrophic approximation is no longer valid in the highly nonlinear phase (t =1.52). Whilethe quasi-geostrophic system gives a good estimation for the maximum growth rates of the linear stabilityanalysis in the shallow water system even for relatively high Fr, the effect of gravity wave cannot benegligible in the highly nonlinear phase. A considerable amount of gravity wave components is radiatedfrom unsteady balanced flows in this parameter range (Sugimoto et al., 2006). Since the phase speeds ofthese radiated gravity waves are no longer much faster than the flow speeds associated with the rotationalcomponents, these gravity waves interact with the rotational components in the highly nonlinear phase.Therefore � and � are no longer merely slave to .

To evaluate the validity of the quasi-geostrophic approximation, we calculate the root mean squareof the relative deviation of the vorticity in the quasi-geostrophic system from that in the shallow watersystem√∫ 3�

�

∫ 2�0

(sh − qg)2 dx dy/∫ 3�

�

∫ 2�0

(sh)2 dx dy, (25)

where sh and qg mean the vorticity in the shallow water system and the quasi-geostrophic system,respectively, and these are integrated in the jet region only. Figs. 14 and 15 show Ro and Fr dependence


Fig. 12. Same as Fig. 11 except for Ro = 100, Fr = 0.01. t = 1.10 (top) and t = 1.52 (bottom). The contour interval is 1 × 10−4in (c).

of the time-evolution of (25) for fixed Fr = 0.01 and Ro = 100, respectively. While initially the relativedeviation of all cases increase exponentially with time ( ∼ 10), the deviation in the cases of Ro = 1.100for Fr = 0.01 saturate around 10−2. As a result, the quasi-geostrophic approximation is still valid in thehighly nonlinear phase of Ro = 1.100 and Fr = 0.01. On the other hand, a significant departure from theshallow water system appears in the highly nonlinear phase of the other cases; the order of the deviationis around unity. These results are consistent with those in the linear stability analysis, since the relativedeviation of the maximum growth rate in the quasi-geostrophic system from that in the shallow watersystem in Fig. 9 becomes larger in the case of smaller Ro for fixed Fr = 0.01, and smaller in the caseof smaller Fr for fixed Ro = 100. Note that in the case of Ro=10,100 for fixed Fr = 0.01, the relativedeviation is so small that the lines are overlapped in Fig. 14.

4. Discussions

4.1. A priori estimation of ‖�′‖/‖�′‖Here, we discuss the results of Fig. 10 in terms of the characteristics of Rossby waves. Since the

parameter range in which the quasi-geostrophic system gives a good estimation for the maximum growthrate of the most unstable modes in the shallow water system is Fr�1 and Fr2/Ro2 �1, the instability inthis parameter range should be the barotropic instability which is explained by the resonance of neutral


Fig. 13. Same as Fig. 12 except for Ro = 100, Fr = 0.5. t = 1.10 (top) and t = 1.52 (bottom). The contour interval is 2 × 10−1in (c).

Fig. 14. Ro dependence of time-evolution of the relative deviation of (25) for fixed Fr = 0.01. Ro = 0.1 (solid line), Ro = 1(dashed line), Ro = 10 (dotted line), and Ro = 100 (dash-dotted line), respectively. New time scale = t · Ro is defined.


Fig. 15. Fr dependence of time-evolution of the relative deviation of (25) for fixed Ro = 100. Fr = 0.01 (solid line), Fr = 0.05(dashed line), Fr = 0.1 (dotted line), and Fr = 0.5 (dash-dotted line), respectively. Time scale is the same as Fig. 14.

Rossby waves. In the present system, we have a neutral Rossby wave mode which has a dispersion relation(see Appendix B for the detailed calculation)

≈ −�0k�f 20 + �0(k2 + l2)

= −�kk2 + l2 + f 20 /�0

, (26)

where � means the effect of latitudinal gradient of the potential vorticity (dq/dy) and (k, l) represent thewavenumber for x and y directions. We can calculate the ratio of the amplitude ‖�′‖ of the divergent flowto the amplitude ‖�′‖ of the rotational flow of the neutral Rossby wave mode as

‖�′‖‖�′‖ =

√√√√ l24 + f 20 k22k24 + �20

(k2 + l2)2k2 − 2�0(k2 + l2)2 + f 20 l22 . (27)

We make a scale analysis of (27). First, from (26), we obtain the scale of as

∼ −U/L3

1/L2 + 1/L2 + 1/L2 · Fr2/Ro2 ∼ U/L. (28)

Here we have taken account of the scale of (k, l) ∼ 1/L and � ∼ d2u/dy2 ∼ U/L2 and Fr2/Ro2>1. Byusing (28), Eq. (27) can be expressed as

‖�′‖‖�′‖ ∼

√1 + 1/Ro2

1 + 1/Fr4 − 1/Fr2 + 1/Ro2

∼√

Ro2 + 1Ro2 + Ro2/Fr4 − Ro2/Fr2 + 1. (29)


Considering Ro>1, Fr>1 and Fr2/Ro>1 in Regime 1 on (29), we have the result of ‖�′‖/‖�′‖ ∝ Fr2/Roas shown in Fig. 10. Similarly, for Ro?1 and Fr>1, Eq. (27) leads to ‖�′‖/‖�′‖ ∝ Fr2 in Regime 2. Inaddition, since Fr>1 in the whole region in which the quasi-geostrophic system gives a good estimationfor the maximum growth rate of the perturbations in the shallow water system, this ratio is always small(‖�′‖/‖�′‖>1).4.2. Scaling analysis for the perturbations

Scaling analysis for the perturbations (�′, �′, �′) in the linearized shallow water equations (11)–(13) isdone to investigate which terms are dominant in each sub-regime obtained in Section 2. The basic stateis scaled by the parameters of the jet as

u∗ = U0u, �∗ = �0 + f BU0�,du∗

dy∗= U0

B

du

dyand

d�∗

dy∗= f U0

d�

dy.

We change these values for the basic state using the nondimensional parameters, which are defined by(7). As for the perturbations, the nondimensionalized variables are rescaled to have the same order O(1)as

(�′, �′, �′) = (U ′L′�†, �′U ′L′�†, H ′�†),t = ′t† and

(�X′

�x,

�X′

�y

)= 1

L′

(�X′

�x†,

�X′

�y†

),

where U ′, L′, H ′, and ′ are scales of velocity, length, free surface displacement, and time, respectively,and X′ represents each dependent variable. All the new perturbation variables with † are O(1). The ratio‖�′‖/‖�′‖ of the divergent flow to the rotational flow is given by �′, and �′>1 as discussed in Section 4.1and shown in Fig. 10. Nondimensional parameters for the perturbations are also defined as

Ro′T ≡1

f ′, Ro′ ≡ U

′

f L′and F ′ ≡ f

2L′2

H ′.

By using these nondimensional parameters, (11)–(13) are rewritten as

Ro′T�∇†2�†

�t†+ Rou�∇

†2�†

�x†− Ro�′ du

dy∇†2�†

− Rod2u

dy2

(��†

�x†+ �′ ��

†

�y†

)+ �′∇†2�† = 0, (30)

Ro′T �′�∇†2�†

�t†+ Ro�′u�∇

†2�†

�x†− 2Rodu

dy

�

�x†

(��†

�x†+ �′ ��

†

�y†

)

− ∇†2�† + 1F ′Ro′

∇†2�† = 0, (31)

Ro′T��†

�t†+ Rou ��

†

�x†+ Ro

2

Fr2F ′Ro′�′∇†2�†

+ RoF ′Ro′{

d�

dy

(��†

�x†+ �′ ��

†

�y†

)+ �′�∇†2�†

}= 0. (32)


Here, we have also assumed the characteristic length scale of the perturbation is approximately the sameas the width of the jet; L′ ∼ O(1).4.3. Validity of the quasi-geostrophic approximation for the perturbations

First, we consider why the quasi-geostrophic system for the maximum growth rate is so good anapproximation in a wide parameter range of Fr>1 and Fr2/Ro2>1, even for high Ro. In this range,the rotational amplitude is much greater than the divergence amplitude as discussed in Section 4.1 andshown in Fig. 10, hence �′>1. Furthermore, �′>Ro, since Fr2/Ro2>1. These two inequalities for �′ yield�′>min(1, Ro). By using this inequality, Eq. (30) can be rewritten approximately as(

Ro′T�

�t†+ Rou �

�x†

)∇†2�† − Rod

2u

dy2��†

�x†≈ 0. (33)

In this range of Fr2/Ro2>1, on the other hand, the linearized quasi-geostrophic equation (15) can beapproximated as(

�

�t+ u �

�x

)∇2�′g −

d2u

dy2��′g�x

≈ 0. (34)

For any time-dependent flow, Ro′T is approximated equal to Ro in (33). In that case, Eq. (33) is identicalto (34). This is why the maximum growth rate of the quasi-geostrophic system is in good agreement withthat of the shallow water system in this parameter range. In addition, Eq. (33) is a closed equation for �†.This is the reason why the spatial structures of eigenfunction �′ for the most unstable modes are almostindependent of Ro and Fr as shown in Figs. 3 and 5. Since Ro′T ∼ Ro, the maximum growth rate mustbe proportional to Ro as seen in Fig. 6. As long as this relation is maintained, the dominant flow in thenonlinear phase is only the rotational flow , thus � and � are slave to . Since the parameter dependenceof �′ is different in each regime, the role of �′ in (30) is different. This is the reason why the relativedeviation in Fig. 9 have two sub-regimes.

4.4. Balance in Regime 1 (geostrophic balance)

Here we consider the dominant balance in Regime 1. In Regime 1, Ro′T ∼ Ro>1 and �′>1, so thatthe last term on the left-hand side of (31) must be O(1), or F ′Ro′ ∼ O(1), i.e.

−∇†2�† + 1F ′Ro′

∇†2�† ≈ 0, (35)which gives the geostrophic balance. In (32), the geostrophic balance both in the basic flow and theperturbation gives

u��†

�x†+ F ′Ro′ d�

dy

��†

�x†≈ 0, (36)

so that the remaining dominant part in (32) becomes

Ro′T��†

�t†+ Ro

2

Fr2F ′Ro′�′∇†2�† ≈ 0, (37)


since Ro2/Fr2?1, and Ro>1. Eq. (37) shows that the divergence of the perturbation, or irrotational flowmakes the time-evolution of �′, and that �′ is 1/4 wavelength out of phase to �′ (Fig. 5; bottom).

4.5. Balance in Regime 2 (cyclostrophic balance with basic shear)

Next, we consider the dominant balance in the Regime 2. In Regime 2, Ro′T ∼ Ro?1, and again �′>1.By using these inequalities, Eq. (31) must become

−2Rodudy

�

�x†

(��†

�x†

)+ 1

F ′Ro′∇†2�† ≈ 0. (38)

This gives another balance which is related to the latitudinal shear of the basic flow. In order to keepboth terms being the same order, we have F ′Ro′ ∼ Ro−1. Since the term du/dy · �2�†/�x†2 can bewritten as �(du/dy · v†)/�x†, the relationship (38) implies that longitudinal difference of the latitudinalvorticity flux is in balance with the Laplacian of the surface displacement �†. By using �′>1, we obtainthe dominant part in (32) as

Ro′T��†

�t†+ Rou ��

†

�x†+ Ro

2

Fr2F ′Ro′�′∇†2�† ≈ 0 (39)

because RoF ′Ro′ ∼ O(1). In contrast to (37) in Regime 1, the time-evolution of �′ is caused not only by�′, but also by the advection due to the basic flow u.

5. Conclusions

Motivated by our previous study in which gravity waves had been radiated from unsteady balanced jetflow, the stability of a zonal jet was investigated to check the validity of the quasi-geostrophic approxima-tion. This is an example to study a limit of validity of balanced models. We did the linear stability analysesand the nonlinear time-evolutions in the f-plane shallow water system, and the stability characteristicswere compared with those obtained in the quasi-geostrophic system for a wide range of Ro and Fr.

In the results of linear stability analysis, the quasi-geostrophic system gives a good estimation for themaximum growth rate of the perturbations in the shallow water system for a wide parameter range ofFr2/Ro2>1 and Fr>1 even for high Ro (Fig. 6). The ratio of the amplitude of the divergent flow ‖�′‖ tothat of the rotational flow ‖�′‖ is small over all parameter range where the quasi-geostrophic system givesa good estimation for the maximum growth rate of the shallow water system. However, the dependenceof the ratio ‖�′‖/‖�′‖ on the two parameters, Ro and Fr, is different between high and low Ro regimes(Fig. 10). In the low Ro regime (Ro < 5), ‖�′‖/‖�′‖ ∝ Fr2/Ro, while in the high Ro regime (Ro > 5),‖�′‖/‖�′‖ ∝ Fr2.

We also examined the nonlinear time-evolutions numerically in both the shallow water system and thequasi-geostrophic system to check the validity of the quasi-geostrophic approximation and investigatedthe dominant balance in the nonlinear phase. In Regime 1 (low Ro), the quasi-geostrophic system is a goodapproximation for the shallow water system even in the highly nonlinear phase. The dominant balanceis the geostrophic balance similarly as in the linear stability analysis. Furthermore, even in Regime 2(high Ro), as long as Fr is small the quasi-geostrophic system is a good approximation for the shallowwater system even in the highly nonlinear phase. The dominant balance is the cyclostrophic balance inwhich intense vorticity regions are associated with the shallower geopotential height. Although the linear


stability analysis in the quasi-geostrophic system gives a good estimation for the maximum growth ratefor relatively high Fr case and small Ro case in the shallow water system, a significant departure from thequasi-geostrophic system can be seen in the highly nonlinear phase because of the surface deviations.

Considering that the instability occurred in the parameter range swept in this paper is due to theresonance of neutral Rossby waves, we made an estimation for characteristics of the unstable modesusing the dispersion relation of the neutral Rossby waves in this system. Using scaling analysis for theneutral Rossby waves, we explained that the ratio ‖�′‖/‖�′‖ should be proportional to Fr2/Ro in the lowRo regime, while it should be proportional to Fr2 in the high Ro regime. In addition, many characteristicsof unstable modes are well understood by the consideration of leading terms in the linearized shallowwater equations by use of smallness of ‖�′‖/‖�′‖. Scaling analysis for the perturbations was doneseparately from the scaling of the basic state. The dominance of the rotational flow explains why thequasi-geostrophic system gives a good estimation for the maximum growth rate, since the vorticityequation in the shallow water system becomes a single equation for �′ after the scaling, which is inagreement with the linearized quasi-geostrophic equation. The structure of the unstable eigenmodes canbe understood from the divergence equation. The geostrophic balance is dominant in the low Ro regime,while the cyclostrophic balance related to the latitudinal shear of the basic zonal flow is dominant in thehigh Ro regime.

Acknowledgments

Numerical experiment was performed with VPP800 and HPC2500 at the Academic Center for Com-puting and Media Studies, Kyoto University, at the Information Technology Center, Nagoya University,and the KDK system of Research Institute for Sustainable Humanosphere (RISH) at Kyoto University asa collaborative research project. GFD-DENNOU Library was used for drawing the figures. ISPACK-0.61was used for numerical experiments and analyses. N. Sugimoto is supported by Grant-in-Aids for the21st Century COE programs “Elucidation of the Active Geosphere” and “Frontiers of ComputationalScience”. The authors thank Prof. S. Kida, Prof. Y.-Y. Hayashi, and three anonymous referees for theirconstructive comments.

Appendix A

The purpose of this appendix is to obtain the eigenmode of a divergence component in the quasi-geostrophic system. In the linearized shallow water equations (8)–(10), we divide the perturbations(u′, v′, �′) into geostrophic components (ug, vg, �g) of O(1) and ageostrophic components (ua, va, �a)of O(Ro),

(u′, v′, �′) = (ug, vg, �g) + Ro(ua, va, �a). (40)The key point of introducing the quasi-geostrophic system is that the time-evolution of the geostrophiccomponents depend on the ageostrophic components at O(Ro), which yields

�ug

�t+ Rou�ug

�x+ Rodu

dyvg − va = −��a

�x, (41)

�vg

�t+ Rou�vg

�x+ ua = −��a

�y, (42)


where the geostrophic balance (ug, vg)=(−��g/�y, ��g/�x) has been used. Taking a rotation of (41)–(42),we have

�g

�t+ Rou�g

�x− Rod

2u

dy2vg = −∇2�a . (43)

Therefore, using g = ∇2�g and vg = ��g/�x, we can obtain the profile of the divergence component �afrom the eigenmode of �g as

�∇2�g�t

+ Rou�∇2�g

�x− Rod

2u

dy2��g

�x= −∇2�a . (44)

Appendix B

The purpose of this appendix is to obtain the dispersion relation for the neutral Rossby waves inthe present system. First, the shallow water equations (1)–(3) are linearized with the static basic state(u, v, �) = (0, 0, 0), which yields

�u

�t= f v − ��

�x, (45)

�v

�t= −f u − ��

�y, (46)

��

�t+ �0

(�u

�x+ �v

�y

)= 0. (47)

Here, taking account of the latitudinal change of Colioris parameter f, we assume wave solution of(x, t) as(

u

v

�

)∝(

U(y)

V (y)

P (y)

)ei(kx−t). (48)

Substituting these wave solutions in (45)–(47), we obtain

iU = f V − ikP , (49)

iV = −f U − dPdy

, (50)

iP + �0(

ikU + dVdy

)= 0. (51)

Thus we have the following equations from (49)–(51):

(2 − �0k2)P = −i�0(

d

dy− kf

)V , (52)

(2 − �0k2)U = −i(

�0kd

dy− f

)V . (53)


Substituting (52) and (53) into (51), we obtain the equation only for V as[�0

d2

dy2+ (2 − f 2) − �0k df

dy− �0k2

]V = 0. (54)

Here, let us introduce �-plane approximation

f = f0, dfdy

= �. (55)

By using (55), we can assume a wave solution for y direction as

V (y) ∝ eily , (56)where l stands for the wavenumber in y direction. Therefore we obtain the eigenvalue equation

3 − {f 20 + �0(k2 + l2)} − ghk� = 0. (57)Eq. (57) has three solutions. One is the following solution for Rossby waves:

≈ −�0k�f 20 + �0(k2 + l2)

= −�kk2 + l2 + f 20 /�0

. (58)

The other two are solutions for gravity waves propagating opposite directions. From (58), the eigenfunc-tion for Rossby waves are given by

v = Aei(kx+ly−t), (59)

u = −i(i�0kl − f0)

2 − �0k2 Ae

i(kx+ly−t), (60)

where A is an arbitrary constant. From (59) and (60), we obtain the ratio ‖�‖/‖�‖ as‖�‖‖�‖ =

∥∥�u/�x + �v/�y∥∥∥∥�v/�x − �u/�y∥∥ =∣∣∣∣k(−i(i�0kl − f0)/(2 − �0k2)) + lk − l(−i(i�0kl − f0)/(2 − �0k2))

∣∣∣∣=√

l24 + f 20 k22k24 + �20(k2 + l2)2k2 − 2�0(k2 + l2)2 + f 20 l22

. (61)

References

Balmforth, N.J., 1999. Shear instability in shallow water. J. Fluid Mech. 387, 97–127.Charney, J.G., 1947. The dynamics of long waves in a baroclinic westerly current. J. Meteorol. 4, 135–163.Ford, R., 1994. Gravity wave radiation from vortex trains in rotating shallow water. J. Fluid Mech. 281, 81–118.Ford, R., McIntyre, W.E., Norton, W.A., 2000. Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57,

1236–1254.Gent, P., McWillams, J.C., 1983a. Consistent balanced models in bounded and periodic domains. Dyn. Atmos. Oceans 7,

67–93.Gent, P., McWillams, J.C., 1983b. Regimes of validity for balanced models. Dyn. Atmos. Oceans 7, 167–183.Hartmann, D.L., 1983. Barotropic instability of the polar night jet stream. J. Atmos. Sci. 40, 817–835.


Hoskins, B.J., 1975. The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32,233–242.

Ishioka, K., 2002. ispack-0.61, 〈http://www.gfd-dennou.org/arch/ispack/〉, GFD Dennou Club.Kubokawa, A., 1985. Instability of a geostrophic front and its energetics. Geophys. Astrophys. Fluid Dyn. 33, 223–257.Orlanski, I., 1968. Instability of frontal waves. J. Atmos. Sci. 25, 178–200.Ripa, P., 1983. General stability conditions for zonal flows in a one-layer model on the �-plane or the sphere. J. Fluid Mech.

126, 463–489.Satomura, T., 1981. An investigation of shear instability in a shallow water. J. Meteor. Soc. Jpn 59, 148–167.Spall, M.A., McWilliams, J.C., 1992. Rotational and gravitational influences on the degree of balance in the shallow-water

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30, 2562–2573.Sugimoto, N., Ishioka, K., Yoden, S., 2006. Gravity wave radiation from unsteady rotational flow on an f -plane shallow water

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http://www.gfd-dennou.org/arch/ispack/

Balance regimes for the stability of a jet in an f-plane shallow water systemIntroductionLinear stability analysisModel descriptionLinearization and eigenvalue analysisResults

Nonlinear time-evolutionModel descriptionResults

DiscussionsA priori estimation of "026B30D phi"026B30D /"026B30D psi"026B30D Scaling analysis for the perturbationsValidity of the quasi-geostrophic approximation for the perturbationsBalance in Regime 1 (geostrophic balance)Balance in Regime 2 (cyclostrophic balance with basic shear)

ConclusionsAcknowledgmentsAppendix A Appendix B References

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