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Boundary layer dissipation and the nonlinear interaction of equatorialbaroclinic and barotropic rossby wavesJoseph A. Biello a;Andrew J. Majda a

a Courant Institute of Mathematical Sciences, New York, NY 10012, USA

To cite this Article Biello, Joseph A. andMajda, Andrew J.(2004) 'Boundary layer dissipation and the nonlinear interactionof equatorial baroclinic and barotropic rossby waves', Geophysical & Astrophysical Fluid Dynamics, 98: 2, 85 — 127To link to this Article: DOI: 10.1080/03091920410001686712URL: http://dx.doi.org/10.1080/03091920410001686712

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Geophysical and Astrophysical Fluid DynamicsVol. 98, No. 2, April 2004, pp. 85–127

BOUNDARY LAYER DISSIPATION AND THE

NONLINEAR INTERACTION OF EQUATORIAL

BAROCLINIC AND BAROTROPIC ROSSBY WAVES

JOSEPH A. BIELLO* and ANDREW J. MAJDA

Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA

(Received 16 July 2003; In final form 5 December 2003)

Two-layer equatorial primitive equations for the free troposphere in the presence of a thin atmosphericboundary layer and thermal dissipation are developed here. An asymptotic theory for the resonant nonlinearinteraction of long equatorial baroclinic and barotropic Rossby waves is derived in the presence of suchdissipation. In this model, a self-consistent asymptotic derivation establishes that boundary layer flows aregenerated by meridional pressure gradients in the lower troposphere and give rise to degenerate equatorialEkman friction. That is to say, the asymptotic model has the property that the dissipation matrix has oneeigenvalue which is nearly zero: therefore the dynamics rapidly dissipates flows with pressure at the base ofthe troposphere and creates barotropic/baroclinic spin up/spin down. The simplified asymptotic equationsfor the amplitudes of the dissipative equatorial barotropic and baroclinic waves are studied by lineartheory and integrated numerically. The results indicate that although the dissipation slightly weakens thetropics to midlatitude connection, strong localized wave packets are nonetheless able to exchange energybetween barotropic and baroclinic waves on intraseasonal timescales in the presence of baroclinic meanshear. Interesting dissipation balanced wave-mean flow states are discovered through numerical simulations.In general, the boundary layer dissipation is very efficient for flows in which the barotropic and barocliniccomponents are of the same sign at the base of the free troposphere whereas the boundary layer dissipationis less efficient for flows whose barotropic and baroclinic components are of opposite sign at the base of thefree troposphere.

Keywords: Equatorial Rossby waves; Atmospheric boundary layer; Amplitude equations

1. INTRODUCTION

The meridional exchange of energy between the midlatitudes and equatorial regions isa topic of considerable significance for understanding global teleconnection patternsfrom the tropics to the midlatitudes. Several authors have used linearized two-layerprimitive equations on a �-plane in order to describe the interaction of long equatorialbaroclinic Rossby waves and long equatorial barotropic Rossby waves. The lineartheory of the primitive equations in the zonal long-wave limit yield two sets ofwaves. The first are weakly dispersive, zonally long, equatorially trapped baroclinic

*Corresponding author. E-mail: [email protected]

ISSN 0309-1929 print: ISSN 1029-0419 online � 2004 Taylor & Francis Ltd

DOI: 10.1080/03091920410001686712

Downloaded By: [New York University] At: 16:45 7 May 2010

Rossby waves. For a dry gravity wave speed of 50ms�1 the fastest of these travel at�16:6ms�1 (see Majda, 2003, for a comprehensive discussion). The primitive equationsalso have linear solutions which correspond to equatorial barotropic Rossby waveswith a significant midlatitude projection. In the long-wave limit these waves, too, aredispersionless and travel westward at a speed proportional to their wavelength squared.

Equatorial baroclinic Rossby waves provide a description of planetary scale pheno-mena in the tropics. Similarly, equatorial barotropic Rossby waves describe phenomenalinked with potentially significant connections between the tropics and midlatitudes.Several authors (see Webster, 1972, 1981, 1982; Kasahara and Silva Dias, 1986;Hoskins and Jin, 1991; Wang and Xie, 1996; Lim and Chang, 1981, 1986; Hoskinsand Yang, 2000) have studied these waves in the context of midlatitude connectionsto the tropics. These studies have found that nearly dispersionless long equatorialbaroclinic Rossby waves can exchange significant energy with barotropic waves atmidlatitudes in the presence of vertical and meridional mean shears.

With the benefit of this intuition Majda and Biello (2003) and Biello and Majda(2003) recently have derived a long-wave theory describing the nonlinear interactionof equatorial baroclinic waves and barotropic waves in the presence of zonal meanbarotropic shears and zonal mean vertical shears. The partial differential equationsof the theory describe waves longer than 5000 km in the zonal direction and 1500 kmmeridionally. The theory admits waves of amplitude on the order of 5ms�1 near theequator interacting with zonal mean velocities and vertical shears of about the samestrength. In these studies it was shown that the presence of vertical, baroclinic shearallows for significant energy exchange between the tropically confined baroclinicwaves and barotropic waves with significant midlatitude projection on timescales onthe order of fifteen days. While detailed comparison with observations of the presenttheories (this work and that of Majda and Biello (2003) and Biello and Majda (2003))will be presented elsewhere, equatorially trapped Rossby waves with a significantbarotropic component are often observed propagating westward in the easternPacific (Kiladis and Wheeler (1995) and Wheeler et al. (2000)).

Important additional effects not considered by Majda and Biello (2003) and Bielloand Majda (2003) are dissipative mechanisms arising from radiative cooling and atmos-pheric boundary layer drag. In this article we shall systematically incorporate thesemechanisms in the asymptotic theory of Majda and Biello (2003) and Biello andMajda (2003) following the resonant long-wave theory that they developed for equator-ial baroclinic and barotropic waves: hereafter we refer to these papers collectively asMB. The models we develop here show that, due to boundary layer dissipation, variousanomalies on timescales of ten days can strengthen upper troposphere flow whilesimultaneously weakening flow at the base of the troposphere. This feature is part ofthe observational record of anomalies such as the intraseasonal oscillation (see Yanaiet al., 2000).

A two-layer tropospheric model with weak Newtonian cooling is coupled to aconvectively well-mixed, thin barotropic boundary layer (see Neelin, 1988; Wang andLi, 1993; Moskowitz and Bretherton, 2000, for the origin of similar boundary layermodels). Following MB, resonantly interacting baroclinic and barotropic troposphericlong waves are studied. Here the mean tropical climatology is represented by fairlyweak (order 5ms�1) barotropic and baroclinic shears which are appropriate forthe tropics. Though the boundary layer dissipation times are relatively small for the(12 days) intraseasonal timescales of interest here, the boundary layer is thin in

86 J.A. BIELLO et al.


comparison to the troposphere. This combination allows for a longer relaxation time inthe troposphere due to boundary layer dissipation and leads in a systematic fashion toequatorial Ekman friction which is described by a degenerate linear operator. Sinceradiative cooling is weak, it is straightforward to incorporate in the higher orderterms of the asymptotic theory of MB.

In Section 2 we describe the basic two-layer model with a well-mixed barotropicboundary layer. In Section 2.1 the long-wave asymptotics is described while in Section2.2 the linear theory of low Froude number flows is described in detail. Section 2.3assembles these pieces into a systematic long-wave theory of equatorial baroclinic andbarotropic Rossby waves in the presence of boundary layer dissipation. In Section 3amplitude equations are constructed for resonantly interacting baroclinic and barotro-pic wavetrains. The new linear dissipation terms are evaluated in Section 3.1 and thenormal form rescaling from MB is provided in Section 3.2. The linear theory ofmean flows and waves is described in Section 4. Section 5 provides a series of numericalexamples which illustrate the nonlinear behavior of the amplitude equations withdissipation included. Solitary waves solutions are used in Section 5.1 to demonstratethe breaking of vertical symmetry by dissipation. Sections 5.2 and 5.3 revisit theexamples of midlatitude to tropics wave energy exchange that were studied by MB.Section 5.4 considers the long-time evolution to a nonlinear balanced state. Weconclude in Section 6 with a brief discussion of our results.

2. ATMOSPHERIC BOUNDARY LAYER AND RADIATIVE DAMPING

EFFECTS ON LONG EQUATORIAL BAROCLINIC AND

BAROTROPIC WAVES

We shall systematically incorporate boundary layer drag and thermal dissipation inthe description of resonantly interacting equatorial baroclinic and barotropic longwaves. To this end we consider a standard two-layer �-plane model describing thefree troposphere, whose vertical extent is 0 � z � HT . In addition, at the base ofthe free troposphere there is assumed to be a well-mixed, barotropic boundary layerwith extent �zB � z � 0. The velocity profile as a function of height is sketched inFig. 1.

The equatorial baroclinic and barotropic waves in the free troposphere are describedby much the same equations previously considered by Majda and Biello (2003), andBiello and Majda (2003). The nonlinear equations for the interaction of barotropic,v0, and baroclinic, v1, modes in the free troposphere can be derived in standard fashion(Neelin and Zeng, 2000; Majda and Shefter, 2001) from a two-vertical mode Galerkintruncation of the Boussinesq equations on the �-plane with rigid lid boundary con-ditions with the form

v ¼ v0ðx, yÞ þ v1ðx, yÞffiffiffi2

pcosð�z=HT Þ,

p ¼ p0ðx, yÞ þ p1ðx, yÞffiffiffi2

pcosð�z=HT Þ,

ð1Þ

where pj are the pressures associated with each mode and HT ¼ 16:75 km is thetroposphere thickness. Using the gravity wave speed, c ¼ HT

�NN=� determined by theBrunt–Vaisala frequency, �NN ¼ 10�2 s�1, we consider the nondimensionalized equations

NONLINEAR EQUATORIAL ROSSBY WAVES 87


with the unit of meridional and zonal length measured by the equatorial Rossby radius,LE ¼ c=�ð Þ

1=2 and time given by TE ¼ c�ð Þ�1=2. Velocity and pressure are nondimensio-

nalized by c and c2, respectively. The natural nondimensional vertical length scale in ismeasured by H ¼ HT=� � 5 km while the units for nondimensional vertical velocity, w,are H /TE. Following MB, the standard values associated with dry wave propagation ofa baroclinic heating mode with

c ¼ 50ms�1, TE ¼ 8:3 h, LE ¼ 1500 km ð2Þ

will be used below to demonstrate various qualitative effects of the nonlinear coupling.Hereafter all functions and variables are nondimensional.

The equations

�DDv0�DtDt

þ y v?0 þ JE v1 � v1ð Þ ¼ �Jp0 �12ðJEv0Þv0, ð3Þ

JE v0 ¼ ��B JE vBð Þ, ð4Þ

(b)

z = zB

z = 0

z = HT

(a)

Troposphere

Boundary Layer

FIGURE 1 The structure of the velocity as a function of depth. (a) The left curve shows the barotropicmean wind in the free troposphere and boundary layer as a function of height; horizontal velocity is notcontinuous across z¼ 0 and there is a vortex sheet. The right curve shows the baroclinic mean shear whichexists only in the free troposphere in this model. The dependence of the vertical velocity on height is shownin (b). In the left curve is plotted the barotropic component which is zero at the ground, increases linearlyto the top of the boundary layer and then decreases linearly to zero throughout the troposphere. Thisvertical velocity is the source of the divergence of the horizontal component of the barotropic flow in thetroposphere. The baroclinic component is plotted at the right and is zero at the top and bottom of the freetroposphere.



�DD v1�DDt

þ y v?1 þ v1EJð Þv0 ¼ �Jp1 �12ðJEv0Þv1, ð5Þ

�DDp1�DDt

þ JE v1 ¼ �dd�p1 �ffiffiffi2

p�B ðJEvBÞ �

12ðJEv0Þp1, ð6Þ

DB vB

Dtþ y v?B ¼ �Jðp0 þ

ffiffiffi2

pp1Þ � dd vB � 1

2vB ðJEvBÞ, ð7Þ

describe a barotropic, v0, and baroclinic, v1, velocity field where vj ¼ ðuj, vjÞ, v?j ¼

ð�vj, ujÞ, pj are functions of the horizontal variable, ðx, yÞ alone and all the vectorderivatives involve only horizontal differentiation. Here and elsewhere in the articlethe transport operator �DD=Dt ¼ @=@tþ v0EJ, represents advection by the barotropicmode. The quantity vB ¼ ðuB, vBÞ represents the velocity in the boundary layer andthe advective derivative in the boundary layer refers to the horizontal boundary layervelocity, DB=Dt ¼ @=@tþ vBEJ. A Newtonian cooling term, dd� p1 appears in Eq. (6)and will be further discussed below.

The boundary layer has thickness zB and the ratio

�B � zB=HT ð8Þ

is the fundamental nondimensional quantity appearing in Eqs. (3)–(6) and we shallexploit the fact that it is a small parameter. The profile of horizontal velocity in dimen-sional variables as a function of height in the troposphere is shown in Fig. 1(a). It isclear that the barotropic mode corresponds to a vertically homogeneous horizontalmean wind in the free troposphere whereas the baroclinic mode yields a net verticalshear. For further discussion of the equations describing the dynamics of the equato-rial baroclinic and barotropic waves the reader is referred to Wang and Xie (1996)and MB.

In this model, the atmospheric boundary layer at the base of the troposphere isassumed to be convectively well-mixed and therefore barotropic. Similar boundarylayer models have been used previously in the context of tropical baroclinic waves(Neelin, 1988; Wang and Li, 1993; Moskowitz and Bretherton, 2000). The boundarylayer has nondimensionalized vertical extent ��B � z � 0, horizontal velocity vBand it is assumed that all of the frictional dissipation occurs there through the termdd vB in (7). Equation (7) and the terms which couple the boundary layer velocity tothe free troposphere in (3)–(6) are derived by assuming that the barotropic boundarylayer is coupled to the troposphere through the continuity of vertical velocity and press-ure across the fixed interface, z ¼ 0. The boundary layer pressure on the right-hand sideof (7), pB ¼ p0 þ

ffiffiffi2

pp1, is simply the sum of the baroclinic and barotropic pressure at

the base of the free troposphere according to (1). A vertical velocity which decreaseslinearly with height is thereby imparted on the barotropic flow in the free troposphereabove the boundary layer which, in turn, is the source of the divergence of v0 in (4).In fact, the nondimensional vertical velocity in the troposphere, w, associated with aboundary layer velocity, vB, and baroclinic velocity, v1 is

w ¼ ��B ð�� zÞJEvB �ffiffiffi2

pJEðv1Þ sin z, 0 � z � � ð9Þ



where the first term in (9) is the barotropic vertical velocity which vanishes at the top ofthe troposphere, z ¼ �, and the second, the baroclinic vertical velocity, vanishes at thetop and bottom of the troposphere. The vertical velocity as a function of height due toeach of these components is shown in Fig. 1(b). Note that as in other boundary layermodels for engineering flows, the horizontal velocity is permitted to jump discontinu-ously across the boundary layer through the boundary conditions which match onlypressure and vertical velocity.

The interface at z¼ 0 has nonzero vertical velocity and should therefore sustaininterfacial waves. Strictly speaking the boundary condition across the interface betweenthe free troposphere and the barotropic boundary layer should be continuity of press-ure and continuity of normal velocity across z ¼ �ðx, y, tÞ where � describes this freeinterface. In addition the evolution of � is described by a kinematic constraint relatingits advective derivative to the vertical velocity at the interface. A Boussinesq modeldescribing a free troposphere interacting with a barotropic boundary layer can bewritten down with these assumptions and, in the absence of dissipation, the non-linear interactions in this model would automatically conserve energy. However, thevertical Galerkin truncation which separates such a model into a barotropic and firstbaroclinic requires a basis which conforms to the dynamic boundary at z ¼ �. Suchan approach has been carried out systematically by Ripa (1995) in a somewhat simplercontext.

Since the boundary layer is both barotropic and thin in comparison to the free tropo-sphere, �B � 1, both the vertical velocity at the interface and the deviation of theinterface from its equilibrium, � are small. Therefore the boundary condition can effec-tively be applied at z¼ 0 with the deviations leading to higher order nonlinearities. Thequadratic nonlinear terms on the right-hand side of (3)–(7) are added in a somewhatad hoc fashion so that nonlinear advection in the absence of dissipation conservesenergy. Therefore they model the effect of the variation of boundary layer thicknesswhich is otherwise not included when assuming a stationary interface. Since the termscontribute only higher order effects in the asymptotic expansions developed here theirspecific form is not relevant for any of the results in this article.

Recall that the ratio of boundary layer thickness to the height of free the troposphereis �B ¼ zB=HT and we will exploit the fact that the boundary layer is thin in compar-ison to the troposphere. In practice, the height of the well-mixed tropical boundarylayer varies from 0:5 to 1:0 km (Wang and Li, 1993; Moskowitz and Bretherton,2000). Thus, the reasonable values of boundary layer thickness and troposphereheight zB ¼ 1:0 km, HT ¼ 16:75 km yield

�B � 0:06, ð10Þ

which we shall use as a representative value for purposes of illustration throughout therest of this article. A boundary layer dissipation timescale can be calculated using aturbulent drag coefficient of CD ¼ 10�3 (Majda and Shefter, 2001), typical turbulentvelocities of 2�5ms�1 and the boundary layer thickness. This yields a dissipationtime of order one day in the boundary layer so that the nondimensional drag coeffi-cient is

dd � 0:3 ð11Þ



with TE � 8:3 h. The model constructed in the following sections explicitly illustratesthat the combination of a thin boundary layer with a one day relaxation time sets alonger boundary induced decay time in the free troposphere in a highly anisotropicfashion.

The effect of thermal dissipation is included through a Newtonian cooling term in thebaroclinic pressure equation (5). Thermal damping is observed to occur on roughly18–20 day timescales yielding

dd� � 0:02 ð12Þ

as a representative value used below.As remarked Eqs. (3–7) conserve energy in the absence of boundary layer drag and

thermal dissipation, otherwise the energy decays. The energy relation can be readilyderived and yields

d E

dt¼

1

2

ZD

jv0j2 þ jv1j

2 þ p21 þ�BjvBj2

� �dx dy

¼ ��B

ZD

jvBj2 dx dy� d�

ZD

p21 dx dy:

ð13Þ

The integrals are taken over the two-dimensional �-plane which is periodic in x(zonally) and bounded in y (meridionally).

2.1. Zonal Long-wave Scaling

Our theory is concerned with tropospheric flows which vary slowly in the zonal direc-tion and evolve on longer timescales than the equatorial time, TE. We shall considerflows whose meridional variations occur on Rossby deformation lengthscales, LE,while zonal variations occur on a longer length L and evolve on timescales, T, whichare long compared to TE. Since �B is small, the barotropic velocity field is primarilysolenoidal with a potentially significant projection at midlatitudes. Furthermore, it iswell known that baroclinic flows in the equatorial wave guide have primarily zonalpropagation with a limited extent of a few thousand kilometers in the north-southdirection but a much larger east-west scale varying over the entire 40 000 km circumfer-ence of the globe (Wang and Xie, 1996; MB). These considerations suggest that weconsider flows whose meridional velocity is small compared to the zonal velocity. Asin MB (also see Section 9.3 of Majda, 2003) define the small parameter, � as follows

LE=L ¼ �, TE=T ¼ �, jvj ¼ �juj, ð14Þ

for �� 1 and seek solutions of the equations in (3) and (5) for the free troposphere ofthe form

pj ¼ pj �x; y; �tð Þ, uj ¼ uj �x; y; �tð Þ, vj ¼ �vj �x; y; �tð Þ ð15Þ

for j ¼ 0, 1.



Boundary layer flow is driven by pressure gradients in the overlying troposphere.In the long-wave approximation for the free troposphere the solutions have meridionalpressure gradients which dominate zonal gradients. In the boundary layer linear theory,the pressure forcing is balanced against the Coriolis effect and boundary layer dissipa-tion. Therefore, for the flow in the boundary layer we also assume meridional variationsover a long-length scale but we do not require meridional velocities to be Oð�Þ smallerthan zonal velocities. Therefore we seek solutions of the boundary layer velocity of theform

vB ¼ vB �x, y, �tð Þ: ð16Þ

It is the divergence of the boundary layer dissipation which couples the boundary flowto the free troposphere baroclinic and barotropic equations. According to the scalinghypothesis in Eq. (16), the meridional and zonal components of the boundary layerflow are of the same order, yet the long-wave scaling means that zonal variations areOð�Þ smaller than meridional variations. Therefore the divergence of the boundarylayer velocity is dominated by the meridional derivative of the meridional boundarylayer velocity; the zonal derivative of the zonal component is Oð�Þ smaller. It willbecome clear that, in this approximation, only the meridional component of the bound-ary layer velocity plays a role in the dissipation of the waves in the free troposphere.These assumptions differ from the ones in Moskowitz and Bretherton (2000) wherelinearized boundary layer equations are simplified through long-wave asymptotics.

With these approximations, the long-wave scaled equations as in MB emerge. Withthe ansatz in (15) and (16) the barotropic and baroclinic equations for the free tropo-sphere from (3) and (5) become

�DDu0�DtDt

� yv0 þ JE u1v1ð Þ þ@p0@x

¼�B

2�

@vB@y

þ �@uB@x

� �� u0

yu0 þ@p0@y

þ �2�DDv0�DtDt

þ JE v1v1ð Þ

� �¼

�B

2

@vB@y

þ �@uB@x

� �� v0

@u0@x

þ@v0@y

¼ ��B

�

@vB@y

þ �@uB@x

� �� ð17Þ

and

�DDu1�DtDt

þ v1EJu0 � yv1 þ@p1@x

¼�B

2�

@vB@y

þ �@uB@x

� �� u1

yu1 þ@p1@y


þ v1EJv0

� �¼

�B

2

@vB@y

þ �@uB@x

� �� v1

�DDp1�DtDt

þ JEv1 ¼�B

�

@vB@y

þ �@uB@x

� �� p1

2�

ffiffiffi2

p� �dd�

�p1

ð18Þ



Finally, the long-wave scaled barotropic boundary layer equations are

�@uB@t

þ@

@xu2B �

þ@

@xp0 þ

ffiffiffi2

pp1

� � �þ@

@yuB vBð Þ ¼ y vB � dd uB

�@vB@t

þ@

@xuB vBð Þ

� �þ@

@yv2B �

þ@

@yp0 þ

ffiffiffi2

pp1

� ¼ �y uB � dd vB:

ð19Þ

Together, these equations are the Long Wave Scaled Equations for EquatorialBaroclinic and Barotropic waves (LWSEBB, see MB) in the presence of thermal andboundary layer dissipation. To avoid cumbersome notation, the arguments of thevariables in (15) and (16) are written implicitly in the LSWEBB and are still denotedby x, y, t.

2.2. Leading Order Asymptotics in a Low Froude Number Expansion

The tropospheric zonal velocities in (17) and (18) are measured in units of the propaga-tion speed of a dry baroclinic mode, 50ms�1 from (2). Typical mean winds and verticalshears at the equator are on the order of 5–10ms�1. As in MB this motivates a smallamplitude solution of the long-wave equations (17)–(19) of order �, where �, theFroude number, is the ratio of typical wind velocities to the dry wave propagation velo-city, c. Combining (15) and (16) with this low Froude number assumption we constructasymptotic solutions of (3), (5) and (7) with the form

pj ¼ �pð1Þj �x; y; �t; ��tð Þ þ �2pð2Þj �x; y; �t; ��tð Þ þ � � � ,

uj ¼ �uð1Þj �x; y; �t; ��tð Þ þ �2uð2Þj �x; y; �t; ��tð Þ þ � � � ,

vj ¼ ��vð1Þj �x; y; �t; ��tð Þ þ ��2vð2Þj �x; y; �t; ��tð Þ þ � � � ,

vB ¼ �vð1ÞB �x; y; �t; ��tð Þ þ �2vð2ÞB �x; y; �t; ��tð Þ þ � � � ,

ð20Þ

where j ¼ 0, 1 as before. A wind velocity of 5ms�1 yields the reasonable value of� ¼ 0:1; we shall fix � ¼ 0:1 in the examples for purposes of illustration as in MB.In the subsequent section a distinguished limit balancing dispersion and nonlinearitywill be sought requiring �2 ¼ �, but for the time being we shall not make such a restric-tion in order to maintain generality. As in MB, the relation �2 ¼ � yields the value� ¼ 0:31 in (15) and (16) as well as for (17)–(19); thus meridional variations are oforder 1500 km while zonal variations are of order ��1LE � 5000 km.

Substituting the scalings of (20) into (17)–(19) the leading order in � and � yields thelinear theory of barotropic and baroclinic waves in addition to the boundary layertheory. Since we are not interested in waves which damp strongly within a wavetravel time we make the mild assumption that

�B=� ¼ o 1ð Þ and dd�=� ¼ o 1ð Þ: ð21Þ



The typical values of �B and dd� given in (10) and (11) fall well within these restrictions.The resultant linear theory describes long equatorial barotropic waves

@u0@t

� yv0 þ@p0@x

¼ 0, yu0 þ@p0@y

¼ 0,@u0@x

þ@v0@y

¼ 0: ð22Þ

and long equatorial baroclinic waves

@u1@t

� yv1 þ@p1@x

¼ 0, yu1 þ@p1@y

¼ 0,@p1@t

þ JEv1 ¼ 0, ð23Þ

each of which are in meridional geostrophic balance and decoupled from the boundarylayer. The equations in (22) and (23) are simply the linear barotropic long-wave equa-tions and the linear equatorial long-wave equation, respectively, discussed by Heckleyand Gill (1984), Majda (2003) and MB.

The linear barotropic long-wave equation (22) has the well-known dispersion rela-tion, !BT ¼ �k=l2, with corresponding dispersionless Rossby wave train solutions,

¼ �BS x� cBTtð Þ sin lyð Þ � BA x� cBT tð Þ cos lyð Þ, cBT ¼ �1=l2 ð24Þ

for any wavenumber l. The velocity components are ðu0, v0Þ ¼ ð� y, xÞ and the super-scripts ‘‘S’’ and ‘‘A’’ denote zonal flows which are symmetric or antisymmetric aboutthe equator, respectively.

Similarly, the linear equatorial long-wave equation, (23), has dispersionless equator-ial Rossby wave train solutions (Heckley and Gill, 1984; Majda, 2003)

p

u

v

0BB@

1CCA ¼

�2�1=2Am x� cmtð Þ Dm�1ðffiffiffi2

pyÞ þDmþ1ð

ffiffiffi2

pyÞ=ðmþ 1Þ

� �2�1=2Am x� cmtð Þ Dm�1ð

ffiffiffi2

pyÞ �Dmþ1ð

ffiffiffi2

pyÞ=ðmþ 1Þ

� �� 2=ð2mþ 1Þ½ �@Am x� cmtð Þ=@x Dmð

ffiffiffi2

pyÞ

2664

3775 ð25Þ

for any integer m > 0 where Dmð�Þ are the parabolic cylinder functions and

cm ¼ �1=ð2mþ 1Þ: ð26Þ

The baroclinic waves have symmetric zonal flows about the equator for m odd and anti-symmetric zonal flows for m even and decay exponentially away from the equator.

From (19) and (20), the leading order boundary layer equations are the equatorialEkman equations

yvB � dduB ¼ 0, yuB þ ddvB ¼ �@

@yðp0 þ

ffiffiffi2

pp1Þ: ð27Þ



These are simply inhomogeneous linear algebraic equations which can be readily solvedin terms of the tropospheric pressure or zonal velocity

uB ¼ ��y

dd2 þ y2

@

@yð p0 þ

ffiffiffi2

pp1Þ ¼

y2 u0 þffiffiffi2

pu1

�dd2 þ y2� ,

vB ¼ �dd

dd2 þ y2

@

@yð p0 þ

ffiffiffi2

pp1Þ ¼

dd y u0 þffiffiffi2

pu1

�dd2 þ y2� :

ð28Þ

The second equalities in (28) arise from the meridional geostrophic balances in (22) and(23) and, using (1), indicate that the boundary layer velocity is proportional to the totalzonal velocity at the base of the free troposphere. In fact, the zonal component of theboundary layer velocity is in the same direction as the total zonal component of thevelocity at the base of the troposphere. On the other hand, the meridional componentof the boundary layer flow points away from the equator (divergent) for westerly windsand toward the equator (convergent) for easterlies. In particular, the meridional bound-ary layer velocity is in the opposite direction of the meridional component of the press-ure gradient (as is the zonal component).

The variation with y of the coefficients in (28) is also interesting. Due to the longzonal scale and long-time approximation and the consequent meridional geostrophicbalance in the troposphere, both the meridional and zonal component of the boundarylayer velocity vanish along the equator, y¼ 0. Furthermore, meridional flows arestronger than zonal flows at the equator yet as jyj ! 1, uB ! ðu0 þ

ffiffiffi2

pu1Þ and the

meridional boundary layer flow tends to zero; the roles of the two components arereversed. Clearly, since both coefficients in Eq. (28) are functions of the parametery=dd the nondimensional parameter, dd, sets the scale for this transition in the relativestrengths in the velocities.

2.3. LWSEBB Equations in the Presence of a Thin Boundary Layer

Under the assumption of weak dissipation, (21), long wavelength linear barotropic andbaroclinic waves are nearly dispersionless and do not dissipate. This suggests that thereis a distinguished limit where the inclusion of a thin barotropic boundary layer withorder one dissipation effectively provides linear dissipation of momentum to the longwaves in the troposphere on a long timescale, � � ��t (refer to (20)). In particular weassume that the leading order barotropic divergence term is Oð�2Þ so that �B=� �;the same will be true of the Newtonian cooling. In order to single out small terms inthe equatorial baroclinic/barotropic equations, we define new order one parameters

�� B

��and �dd� �

dd�

��, ð29Þ

which describe the ratio of the boundary layer thickness to troposphere height and thesmall thermal dissipation coefficient.

In order to have weak dispersive effects compete with nonlinearity in these long-wavesolutions (as in Boyd, 1980; Patoine and Warn, 1982; Majda and Biello, 2003) the zonal



long-wave parameter, �, in (17)–(19) and the amplitude, �, in (20) will be balanced andsatisfy �2 ¼ �. MB discussed how the reasonable value of � ¼ 0:1 allows zonal velocitiesof order 5ms�1 and, with �2 ¼ �, zonal variation on scales of order LE �

�1 � 5000 km;thus the relation �2 ¼ � is physically realistic. With this choice of small parameters, oneunit of the long timescale, �, equals 11 days.

Using the physically relevant values from MB discussed earlier, � ¼ 0:1, � ¼ffiffiffi�

p�

0:31, and the boundary layer thickness and thermal dissipation from (10) and (11),�B � 0:06, dd� � 0:02 the rescaled parameters take on the values

�� 2 and �dd� � 0:67, ð30Þ

both of which are order one so the assumptions in (29) are reasonable.We have assembled all the pieces to construct solutions of the equatorial barotropic/

baroclinic equations in the free troposphere in the presence of the dissipation from theboundary layer and thermal dissipation. Using the scalings described in (20) and (29),the multiple timescales t (which is the � rescaled original time), the slower timescale� ¼ ��t needed in Section 3 while including only the leading order effects from theboundary layer, the long-wave equatorial barotropic equations become

�DDu0�DtDt

� yv0 þ JE u1v1ð Þ þ@p0@x

þ �@u0@�

¼ Oð�3Þ,

yu0 þ@p0@y


þ JE v1v1ð Þ

� �þ �

@v0@�

¼ Oð��3Þ,

@u0@x

þ@v0@y

¼ � ��@vB@y

þOð�2Þ

ð31Þ

and the long-wave equatorial baroclinic equations become

�DDu1�DtDt

� yv1 þ v1EJu0 þ@p1@x

þ �@u1@�

¼ Oð�3Þ,

yu1 þ@p1@y


þ v1EJv0

� �þ �

@v1@�

¼ Oð��3Þ,

�DDp1�DtDt

þ JEv1 þ@p1@�

¼ �� dd�p1 þffiffiffi2

p��@vB@y

� �þOð�3Þ:

ð32Þ

Here the low Froude number scaling in (20) has been used to evaluate the order of theneglected terms. Note that a power of � has been canceled in the first and last equationsin (31) and (32). Also, as in (17)–(19), the convention that the zonal coordinate, x, andtime, t, are the original variables rescaled by � and that the fields have order onevariation on these scales is employed. The meridional boundary layer velocity, vB, isgiven as a function of the baroclinic and barotropic velocities by (28), thus closing(31) and (32).

Since the barotropic velocity is solenoidal at leading order, it is convenient to use astream function and a potential through the Helmholtz decomposition

v0 ¼ J? ð�x; y; �t; ��tÞ þ J�ð�x; y; �t; ��tÞ: ð33Þ



Without canceling �s the nearly incompressible constraint on the velocity field from(3) is

JEv0 ¼ �� @vB@y

ð34Þ

and substituting (33) into (34) yields

�2@2�

@x2þ@2�

@y2¼ � ��

@vB@y

ð35Þ

again the zonal variable, x refers to the long-zonal scale. Notice that zonal potentialflows are order � smaller than meridional flows and, furthermore, @�=@y is order ��smaller than vB so to lowest order the meridional velocity due to potential flow isgiven by

@�

@y¼ � �� vB: ð36Þ

This equality expresses the fact vertically integrated meridional mass flux in the bound-ary layer is balanced by an opposite flux in the troposphere. We can now recast (31)using the stream function. Recognize that, since the velocity from the potential isalready order �� smaller than the other terms, we need only consider its lowest order,linear contribution. The lowest order contribution of the potential terms enters atorder � and arises from the linear term in the velocity equation associated with the�-effect. Therefore taking the curl of the first two equations in (31) yields the barotropicvorticity equation

�DD

�DtDt

@2

@y2þ@

@x� JE

@

@yðv1u1Þ

� �þ �2

�DD

�@t@t

@2

@x2þ JE

@

@xðv1v1Þ

� �� þ � yy;�

¼ ��@

@yð yvBÞ þ Oð�3; �2�2Þ

ð37Þ

The boundary layer zonal velocity is given explicitly in terms of the pressure or tropo-sphere velocity through Eq. (28). The Eqs. (32) and (37) constitute the Dissipative LongWave Scaled Equatorial Baroclinic Barotropic equations (DLWSEBB).

3. AMPLITUDE EQUATIONS FOR DLWSEBB

Amplitude equations for the undamped LWSEBB have been derived and discussed indetail by Majda and Biello (2003) and Biello and Majda (2003). We shall not reproducethe derivations here but instead outline the procedure and focus on the derivation of thenew terms in the amplitude equations.

Using the distinguished limit of dispersion balanced nonlinearity, we set �2 ¼ � forthe remainder of this discussion. The asymptotic series of (20) is substituted into the



dissipative long-wave scaled equatorial baroclinic/barotropic equations (32) and (37)and the resultant equations are solved at each order in �. At lowest order, �, the linearbarotropic long-wave equation (22) and the linear equatorial long-wave equation (22)emerge. As already discussed, their solutions are the dispersionless barotropic Rossbywave trains of (24) and equatorial baroclinic Rossby wave trains of (25).

Only resonantly interacting long waves will yield nontrivial solvability conditions forthe amplitude equations. For a fixed m, the dispersionless packets in (24) and (25) areresonant for a barotropic mode with meridional wavelength, L with l ¼ ð2�=LÞLE

provided that

cBT ðLÞ ¼ cm ¼) l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mþ 1

p, or

L

LE� L ¼

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mþ 1

p ð38Þ

for m ¼ 1, 2, . . .. With the gravity wave speed, c ¼ 50ms�1 and LE ¼ 1500 km from (2)the dimensional meridional wavelengths of the resonant barotropic Rossby waves areapproximately

L ffi 3:6LE ¼ 5440 km for m ¼ 1,

L ffi 2:8LE ¼ 4200 km for m ¼ 2:ð39Þ

The wave packets are referred to a frame moving at the resonant wave speed, �16:7 and�10 ms�1 for m ¼ 1, 2, respectively. Fixing the resonant values of l we proceed tosecond order.

As in MB, the ultimate goal here is to derive simplified dynamics for the interactionof barotropic and equatorially trapped baroclinic waves in spherical geometry. Belowwe make several approximations in deriving the asymptotic models. First, sphericalgeometry is replaced by an equatorial �-plane geometry and, as in MB, the flows onthe �-plane are assumed to be confined to a finite channel domain defined by no pene-tration boundary conditions at y ¼ �L, i.e.

vjy¼�L¼ 0: ð40Þ

Wang and Xie (1996) have established the important fact that linear theory for coupledequatorial baroclinic/barotropic Rossby waves on the �-plane, in the long-wave regimewith baroclinic and barotropic mean shears is an excellent approximation to similarwaves in spherical geometry with boundaries at y ¼ �L where L � 4LE . The valuesof L in (39) are compatible with this. Also, Longuet-Higgins (1968) has shown thatthe �-plane approximation is an excellent one for equatorially trapped baroclinicwaves on the sphere.

Note that as in MB, the symmetric barotropic Rossby mode with amplitude BS in(24) exactly satisfies the boundary condition in (40). The equatorial Rossby wavetrains in (25) do not exactly satisfy these boundary conditions at y ¼ �L but do sowithing exponentially small errors since the dominant contribution from the eigenfunc-tions involve parabolic cylinder functions, Dm(y), with the decay factor, expð�y2=2Þ; fory ¼ L from (39), expð�y2=2Þ � 0:0015, 0:019 for m ¼ 1, 2, respectively. Thus, whilestrictly speaking one should use the more cumbersome exact finite channel equatorialbaroclinic eigenfunctions to exactly satisfy (40), for simplicity in discussion we use



instead the simplified formulas in (25) for the equatorial Rossby waves. For additionalreasons this is an especially good approximation because, according to (20) the meridio-nal flows are already scaled to be weaker in magnitude. As a check, note that the freetropospheric flows in the figures in MB and presented below in the present article allvisually satisfy (40) at y ¼ �L even though the approximation in (25) is utilized.Also, as in the formulas computed in (3.10) and (3.12) from MB, the integrals in theasymptotic interaction coefficients between barotropic and baroclinic waves includingthe boundary layer are extended from the domain, jyj � L, to the entire integrationrange, �1 < y <1. Thus, the interaction coefficients in the asymptotic model arecalculated approximately in this fashion. As regards from MB, this last approximationis a negligible one for the basic interaction coefficients since the ambiguous error termsfrom (3.10) and (3.12) of MB all involve squares of parabolic cylinder functions inte-grated against oscillating periodic functions for jyj > L and, as shown above, theGaussians are already small in amplitude at jyj ¼ L. Here similar approximationsare also used for the new boundary layer contributions. In addition, the leadingorder Ekman flows, �y, in Eq. (36) are picked self-consistently from Eq. (28) and donot satisfy the boundary condition in Eq. (40) at y ¼ �L, but only in the limit,jyj ! 1. Finally, the antisymmetric Rossby wave contribution, BA in Eq. (24) hasbeen carried along below to discuss the symmetries of the interaction throughEkman friction even though it solves stress-free boundary conditions rather thanno-normal flow at y ¼ �L from Eq. (40).

At second order in � (using (20) and �2 ¼ �) the equation governing the stream func-tion from (37) and the baroclinic velocity and pressure from (32) are the linear long-wave equations in (22) and (23) with inhomogeneities arising from the first-orderterms. These inhomogeneous terms are simply functions of the first-order wave packetsand, since they travel at the baroclinic long-wave speed, are resonant with the linearoperator. The resonance gives rise to secular growth of the second-order fields andmust be removed by projection of the nonlinear terms onto the adjoint eigenfunctionsof the linear problem. Since the linear equations are self-adjoint, the adjoint eigenfunc-tions are given by the meridional structure of the baroclinic wave packet in (25) andeach of the meridional structures of the barotropic wave packet from (24). As discussedin detail in MB (and generalized in Biello and Majda (2003)), multiplying the second-order equations by each of these adjoint eigenfunctions and integrating in y results inthe amplitude equations

rA@A

@��DA

@3A

@x3þ

@

@xðABSÞ ¼ �BC,

rB@BS

@x3�DB

@3BS

@x3þ A

@A

@x¼ �S,

rB@BA

@��DB

@3BA

@x3¼ �A

ð41Þ

for each value of m ¼ 1, 2, . . .. The amplitudes Aðx� cmt, �Þ, BSðx� cmt, �Þ and BAðx�

cmt, �Þ with � ¼ �t are defined in (25) and (24) for packets of barotropic and equatorialbaroclinic Rossby waves under the conditions in (38). The coefficients � arise from the



dissipation terms on the right-hand sides of (32) and (37) and are given by

�BC ¼ �

Z 1

�1

pp �dd� pð1Þ1 þ

ffiffiffi2

p��@vð1ÞB

@y

!dy,

�S ¼ � ��

Z L

�L

l cos lyð Þ y v1ð Þ

B dy,

�A ¼ ��

Z L

�L

l sin lyð Þ y v1ð Þ

B dy,

ð42Þ

where pp is the meridional structure of the pressure component of the baroclinic waves,the first entry in (25),

pp ¼ �1ffiffiffi2

p Dm�1 þDmþ1

ðmþ 1Þ

� �ð43Þ

and the meridional flow in the boundary layer is given explicitly in terms of the zonalflow at the base of the troposphere by

vB ¼dd y ðu0 þ

ffiffiffi2

pu1Þ

ðdd2 þ y2Þ: ð44Þ

MB derive amplitude equations which are those of (41) without any dissipationterms, �. The specific forms of the coefficients on the left-hand side of (41) can befound there. We remark that the form of the asymptotic expansion allows for zonalmean barotropic and baroclinic shears which have the same meridional structure asthe waves; simply the zonal means of A, BS and BA. Since waves of different meridionalstructure (i.e. different m) are unable to resonantly interact, one need only consider (41)for a fixed m in isolation from the other modes.

Biello and Majda (2003) consider the more general case when the background shearhas arbitrary meridional structure, in particular a meridional structure opposite to thatof the underlying waves. To fix ideas, let us consider a baroclinic mean zonal shearcomposed of the meridional eigenfunctions of the m¼ 1 and m¼ 2 baroclinic wavesand a zonal mean barotropic shear with meridional structure consisting of l ¼

ffiffiffi3

p

and l ¼ffiffiffi5

psines and cosines. In such instances the equations derived by MB allow

for the exchange of wave energy to l ¼ffiffiffi3

pantisymmetric barotropic waves from

symmetric barotropic waves and baroclinic waves through coupling with the l ¼ffiffiffi5

p

barotropic and m¼ 2 baroclinic mean shears. Similarly l ¼ffiffiffi5

pantisymmetric baro-

tropic waves resonantly exchange wave energy by coupling through the l ¼ffiffiffi3

pzonal

mean barotropic and m¼ 1 mean baroclinic shears. The amplitude equations derivedby MB conserve the zonal mean flows and thus waves of different meridional structure(i.e. m ¼ 1, 2, . . .) can be considered in isolation from the rest. However, the presence ofdissipation allows the evolution – in particular the decay – of the zonal mean shears.A complete description of the waves in this case would require both the m¼ 1 andm¼ 2 equations derived in MB with the addition of the dissipation terms, �. It is astraightforward exercise to add the dissipation terms to the equations considered by



Biello and Majda (2003) and allow the amplitudes of the mean shears to evolve underdissipation. This level of generality is not exploited here but it is interesting for potentialapplications.

3.1. Evaluation of the Dissipation Coefficients

In this section we show that the component of the dissipation vector , arising from theboundary layer has the structure of a nearly degenerate symmetric matrix acting on thevector of wave amplitudes

, ¼

�BC

�S

�A

2664

3775 ¼ � ��

11 12 13

12 22 0

13 0 33

2664

3775

A

BS

BA

2664

3775: ð45Þ

We point out that physically the degeneracy means that there exists a combination ofbarotropic and baroclinic velocities such that

vB ¼ 0¼) u0 þffiffiffi2

pu1 ¼ 0 ð46Þ

is satisfied exactly in the DLWSEBB. However, the amplitude equation (41) project thebaroclinic and barotropic flows onto meridional eigenfunctions such that (46) cannotbe satisfied exactly for all y. A full description of the boundary layer dissipationwould require the complete bases to describe u0 and u1 in which case the dissipationmatrix in (45) would be infinite and exactly degenerate (with an infinite null space).The matrix in (45) is simply a truncation of the infinite matrix and as such retainsonly a near degeneracy.

The component of the dissipation arising from the thermal dissipation only acts onthe baroclinic waves and has the form

�ThermalBC ¼ � �dd��A: ð47Þ

It is straightforward to calculate the thermal dissipation coefficient from (42)

� ¼

Z 1

�1

pp2 dy: ð48Þ

In order to calculate the coefficients of the boundary layer dissipation matrix it is con-venient to define the kernel arising from the Ekman pumping

Fðy; ddÞ � ddð y=ddÞ2

1þ ð y=ddÞ2: ð49Þ



After integrating by parts, using the geostrophic balance from (23), @pp=@y ¼ �yuu, andthe definition of the baroclinic meridional eigenfunction from (25)

uu ¼1ffiffiffi2

p Dm�1ðffiffiffi2

pyÞ �

Dmþ1ðffiffiffi2

pyÞ

ðmþ 1Þ

" #, ð50Þ

it is straightforward to show that the dissipation matrix, !! ¼ ij, from (45) is symmetricand does not couple the two barotropic components. The explicit values of the coeffi-cients are

11 ¼ 2

Z 1

�1

Fðy; ddÞ uu2 dy, 12 ¼ffiffiffi2

pZ 1

�1

Fðy; ddÞ l cosðlyÞ uu dy,

13 ¼ �ffiffiffi2

pZ 1

�1

Fðy; ddÞ l sinðlyÞ uu dy, 22 ¼

Z L

�L

Fðy; ddÞ½l cosðlyÞ�2dy,

33 ¼

Z L

�L

Fðy; ddÞ½l sinðlyÞ�2 dy

ð51Þ

We remark that, strictly speaking, the domain of integration in the barotropicdissipation terms should be taken over �L< y<L. However, since the component ofmeridional boundary layer velocity due to the baroclinic flow in the tropospheredecays rapidly away from the origin, we extend these integrals to infinity for simplicityand at the expense of a small error in the dissipation terms, as done elsewhere in MB.

Further simplification of the matrix !! can be achieved by exploiting the symmetry ofthe zonal component of the baroclinic eigenfunction. In particular it is straightforwardto show that

m ¼ 1 ) uu symmetric ) 13 ¼ 0,

m ¼ 2 ) uu antisymmetric ) 12 ¼ 0:ð52Þ

3.2. Normal form Rescaling

The amplitude equations for the dissipative equatorial baroclinic barotropic waves (41)can be recast in a normal form. Following MB, we define the rescaled amplitudesAA ¼

ffiffiffiffiffirA

pA, BBS ¼ sB

ffiffiffiffiffirB

pBS, BBA ¼ sB

ffiffiffiffiffirB

pBA, the rescaled time �� ¼ �=�0 and zonal coor-

dinate xx ¼ x=x0. Upon removing the hats the dissipative rescaled equatorial baroclinicbarotropic equations (DREBB) are

@A

@��D

@3A

@x3þ@

@xðABSÞ ¼ � ��ð11Aþ 12B

S þ 13BAÞ � �dd��A,

@BS

@x�@3BS

@x3þ A

@A

@x¼ � ��ð12Aþ 22B

SÞ,

@BA

@��@3BA

@x3¼ � ��ð13Aþ 33B

AÞ:

ð53Þ



The values of the rescaling coefficients are presented in the appendix to MB (2003)

ðffiffiffiffiffirA

p,ffiffiffiffiffirB

p, x0, �0, sBÞ ¼ ð1:63, 2:33, 0:60, 1:96, 1Þ for m ¼ 1,

ð1:72, 2:65, 0:55, 4:22, � 1Þ for m ¼ 2:ð54Þ

Using the boundary layer dissipation rate dd ¼ 0:67 the values for the rescaled dissipa-tion coefficients become

� ��0rA

¼0:98,

2:11,

�11 �

11�0rA

¼0:29,

1:05,

�

12 �sB12�0ffiffiffiffiffiffiffiffiffiffirA rB

p ¼0:29,

0,

�13 �

sB13�0ffiffiffiffiffiffiffiffiffiffirA rB

p ¼0,

1:11,

�

22 �22�0rB

¼0:40,

0:79,

�33 �

33�0rB

¼0:50,

1:01,

� ð55Þ

where the first values correspond to m¼ 1 and the second values to m¼ 2.The energy exchange properties of (53) have been studied by MB for the case when

�� ¼ �dd� ¼ 0. We remark that the energy is simply the integral of the amplitudes added inquadrature

E ¼ 12

Z½A2 þ ðBSÞ

2þ ðBAÞ

2� dx, ð56Þ

while each of the wave components of energy are

EA ¼ 12

ZðA0Þ

2 dx, ð57Þ

where the wave components are defined implicitly by

A ¼ �AAþ A0 ¼

ZAdxþ A0 ð58Þ

and similarly for BS and BA. The effects of the dissipation terms will be elucidated in thecoming sections when linear and nonlinear solutions are studied. Note that, from (52)when m¼ 1, antisymmetric barotropic waves decouple from the other modes andsimply disperse and decay. Conversely when m¼ 2 the dissipation does not couple thesymmetric barotropic waves to the (antisymmetric) baroclinic waves. Their couplingoccurs through the original, energy conservative terms on the right-hand side.Dissipative coupling between the antisymmetric baroclinic waves and the antisymmetricbarotropic waves does, however, occur when m¼ 2.



4. LINEAR THEORY OF THE DREBB

For the purposes of discussion, we fix the values of the boundary thickness ratio �� ¼ 2and, when considering thermal dissipation its rate shall be �dd� ¼ 0:67 as discussedabove (30).

4.1. Dissipation of Zonal Mean Flows

Taking the zonal mean of the (53) we find the first novel feature of this model, thatzonal mean flows evolve linearly through a nearly degenerate dissipation matrix. Form¼ 1 with no thermal dissipation the mean flow equations are

�AA�

�BBS�

�BBA�

2664

3775 ¼ �

0:6 0:6 0

0:6 0:8 0

0 0 1

264

375

�AA

�BBS

�BBA

264

375: ð59Þ

This matrix has eigenvalues

�1 ¼ �0:09, �2 ¼ �1:31, �3 ¼ �1:0 ð60Þ

and recognizing that the rescaled time unit corresponds to 22 days we find decay times,Ti ¼ j��1

i j of

T1 ¼ 244 days, T2 ¼ 17 days; T3 ¼ 22 days: ð61Þ

The third eigenvalue clearly corresponds to a pure antisymmetric barotropic mode. Thefirst two eigenvalues correspond to mixed symmetric barotropic and m¼ 1, symmetricbaroclinic mean flows. The very long timescale associated with T1 is a reflection of thenearly degenerate nature of the matrix. Therefore mean flows will relatively quicklydecay to the eigenfunction associated with the smallest eigenvalue, �1, which we callthe slow or slaved state. Since the timescale associated with the smallest eigenvectoris well beyond the time at which other physical effects are expected to intervene, theboundary layer dissipation matrix behaves as though it were degenerate in the asymp-totic equations. The mixed baroclinic/barotropic eigenfunction corresponding to T2 isthe most rapidly dissipated mean state and we will refer to it as the fast eigenfunctionor adjustment flow.

The meridional structure of the mean zonal flow can be reconstructed using theeigenvectors for the slow and fast modes

�BBS � �0:85 �AA, �BBS � 1:18 �AA ð62Þ

respectively. The expression for the total mean zonal flow as a function of the rescaledamplitudes is

u ¼�BSBSð�ÞffiffiffiffiffirB

p l cosðl yÞ ��AAð�ÞffiffiffiffiffirA

p expð�12y2Þ

3

2� y2

� �, ð63Þ



where l ¼ffiffiffi3

pand the positive and negative signs correspond to the bottom and the top

of the free troposphere, respectively. The meridional structure of zonal flow at the topand bottom of the troposphere and the meridional and zonal component of the bound-ary layer flow are plotted in Figs. 2 and 3. Using a barotropic shear which attains avelocity of 3.7ms�1 at the equator, Fig. 2 displays the slaved flow and Fig. 3 displaysthe adjustment flow.

Figure 2(b) shows the zonal flow at the base of the troposphere for the slaved flow.Clearly in the slaved state there is only a weak-zonal flow (about �1ms�1) at the base

−10 0 10

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

(a)

Latit

ude

(km

)

−10 0 10

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

(b)

Velocity (m/s)

−5 0 5

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

(c)

−5 0 5

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

(d)

FIGURE 2 Zonal mean velocity fields for slaved flows, m¼ 1. (a) Zonal mean velocity at the top of the freetroposphere. (b) Zonal mean velocity at the base of the free troposphere. (c) Meridional velocity in theboundary layer. (d) Zonal velocity in the boundary layer.



of the free troposphere near the equator. The meridional velocity in the boundary layer(Fig. 2c), never rises above 1ms�1. In contrast, the zonal flow for the adjustment flow atthe base of the troposphere near the equator is about 9ms�1 in Fig. 3(b). The meridio-nal boundary layer flow is also much larger – about 4ms�1 in Fig. 3(c) – and of theopposite sign of its counterpart in Fig. 2. In both of these cases, the zonal flow inthe boundary layer (Figs. 2 and 3d) near the equator is large if the corresponding mer-idional flow is large. Similarly the zonal flow at the top of the troposphere near theequator (Fig. 2 and 3a) is large if its counterpart at the base of the troposphere issmall. These results clearly reflect the physical principle that boundary layer dissipation

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FIGURE 3 Zonal mean velocity fields for the adjustment flow, m¼ 1. (a) Zonal mean velocity at the top ofthe free troposphere. (b) Zonal mean velocity at the base of the free troposphere. (c) Meridional velocity in theboundary layer. (d) Zonal velocity in the boundary layer.



tends to minimize the meridional (and thereby zonal) velocity in the boundary layernear the equator. Therefore, boundary layer dissipation tends to drive zonal meanflows to a minimum velocity at the base of the troposphere.

For m¼ 2 the zonal mean flows evolve according to

�AA�

�BBS�

�BBA�

2664

3775 ¼ �

2:1 0 2:22

0 1:58 0

2:22 0 2:02

264

375

�AA

�BBS

�BBA

264

375, ð64Þ

where, in this case, one unit of the long timescale corresponds to 46 days. The eigen-values of the dissipation matrix are

�1 ¼ 0:16, �2 ¼ �4:28, �3 ¼ �1:58 ð65Þ

so that the first eigenvalue now corresponds to a growing mode. The timescales associ-ated with these eigenvalues are

T1 ¼ 288 days, T2 ¼ 11 days, T3 ¼ 29 days: ð66Þ

In this case the third eigenvalue represents a decaying symmetric barotropic mean wind.The first eigenvalue yields a small growth of mixed antisymmetric barotropic and anti-symmetric, m¼ 2 baroclinic mean flow. The fact that one mode grows does not posesignificant problems for the theory as again, it happens on timescales much largerthan those of physical interest. In this case, we would expect smaller scale barotropicand baroclinic modes to be spun up prior to the T1 timescale since, as stated before(46) a full description of the boundary layer dissipation would require a completebasis for the antisymmetric baroclinic and barotropic modes. Including only a fewmore baroclinic and barotropic modes actually destroys the positive eigenvalue, butsince these modes do not resonate with the antisymmetric, �10ms�1 baroclinic waveand only interact linearly. Finally, the second mixed mode decays on timescales of11 days and so we can expect a balanced mean flow state corresponding to the firsteigenvector to prevail over long timescales.

Again the meridional structure of the mean zonal flow is reconstructed using theeigenvectors for the slow and fast modes

�BBA � �1:02 �AA, �BBA � 0:98 �AA ð67Þ

respectively. When m¼ 2 the total mean zonal flow is antisymmetric about the equatorand is given by

u ¼�BBAð�ÞffiffiffiffiffirB

p l sinðl yÞ � 2ffiffiffi2

p �AAð�ÞffiffiffiffiffirA

p expð�12y2Þ y�

y3

3

� �, ð68Þ

where l ¼ffiffiffi5

pand the positive and negative signs correspond to the bottom and the top

of the troposphere, respectively. The slaved zonal flow is plotted in Fig. 4 and theadjustment flow in Fig. 5. For the slaved flow, it is again clear that zonal flow is



small at the bottom of the troposphere, especially near the equator. The meridionalflow in the boundary layer is absolutely negligible in this case. Conversely, the adjust-ment flow has exceedingly strong zonal flow at the base of the troposphere and corre-spondingly strong flows in the boundary layer.

We conclude that the effect of the near degeneracy of the boundary layer dissipationmatrix is to quickly drive mean zonal flows to a slow eigenfunction, the slaved meanflow state. The slaved state has minimal flow in the boundary layer and therefore avery weak wind at the base of the troposphere. Since the barotropic and barocliniccomponents of the velocity essentially cancel at the base of the troposphere, they are

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FIGURE 4 Zonal mean velocity fields for the slaved flow, m¼ 2. (a) Zonal mean velocity at the top of thefree troposphere. (b) Zonal mean velocity at the base of the free troposphere. (c) Meridional velocity in theboundary layer. (d) Zonal velocity in the boundary layer.



additive at the top of the troposphere. Therefore the slaved state has very strong windsat the top of the troposphere.

4.2. Breaking the Degeneracy: The Effect of Thermal Dissipation on Mean Flows

Increasing the thermal dissipation has the effect of breaking the near degeneracy due topure boundary layer dissipation. Using m¼ 1 we calculate the eigenvalues and eigen-vectors of the dissipation matrix as a function of dd�. Figure 6(a) shows the dissipationtimes for increasing dd� among a range of reasonable values. The timescale of the slow

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FIGURE 5 Zonal mean velocity fields for the adjustment flow, m¼ 2. (a) Zonal mean velocity at the top ofthe free troposphere. (b) Zonal mean velocity at the base of the free troposphere. (c) Meridional velocity in theboundary layer. (d) Zonal velocity in the boundary layer.



mode rapidly decreases from the 244 day time associated with purely boundary layerdissipation. Nonetheless, for all reasonable values, dd� < 0:67 the slow timescale remainsabove 50 days, well separated from the fast timescale. The latter, in turn, is only mod-erately affected by thermal damping, remaining always on the order of 15 days.

In Fig. 6(b) is plotted the ratio of the zonal velocities at the bottom of the tropo-sphere at the equator for the eigenfunctions of the dissipation matrix as a functionof dd�. Clearly, even when dd� ¼ 0 the wind speed of the slow mode (the negativebranch) does not vanish there; as we have discussed, the zonal velocity of the slowmode becomes low over a broad meridional range. Increasing thermal dampingincreases the barotropic contribution to the slow mode while decreasing its contri-bution to the fast mode.

4.3. Linear Theory of Dissipated Waves for Slaved Mean Flows

Since mean flows are not constant in time, the linear theory of barotropic and barocli-nic waves in the presence of mean flows is not well defined. However, it is useful as avery relevant approximation to consider the linear theory of waves in the presence ofslaved mean flows assuming that the mean velocities remain constant. We describethe linear theory for four cases of m¼ 1 waves interacting with mean shears whichwere chosen to reside in the slaved state. The first two cases have no thermal dissipationwhile the latter two cases use dd� ¼ 0:67; only the two examples with the most extremebehavior are shown in Figs. 7 and 8. The mean shears were chosen to have moderatevalues that can be expected from the climate record so the barotropic mean flowtakes on both positive and negative values. Since the amplitude equations are validonly for the longest wavelengths in the domain we only consider linear theory for thefirst six zonal wavenumbers.

In case 1 the equatorial barotropic mean shear is 2.5ms�1 at the bottom of the tropo-sphere so that the baroclinic mean shear is �3.6ms�1 there. The dissipation time asa function of integer zonal wavenumber is plotted in Fig. 7(a). It is clear that the

0

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ampi

ng ti

me

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s)

0.2 0.4 0.6 0.8 1Thermal dissipation coefficient –1.5

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1

Am

plitu

de r

atio

0.2 0.4 0.6 0.8 1Thermal dissipation coefficient

FIGURE 6 (a) Damping times as a function of thermal dissipation coefficient, dd� for m¼ 1. As dd� increasesthe timescale of the slow mode decreases dramatically, thereby destroying the exact slaving. However, the twodecay timescales do remain well separated. The fast mode is essentially unaffected. (b) The ratio of the zonalmean barotropic to baroclinic velocity at the bottom of the free troposphere at the equator for each of theeigenvectors of the dissipation matrix. The negative branch corresponds to the slowly damped mode.



dissipation times for both the slow and fast modes remain well separated and do notchange significantly over this range of wavenumbers. MB studied the effect of barotro-pic and baroclinic shears on the wave speeds and found that the mean shears couldproduce a significant change in wave speeds, on the order of 5ms�1. Plotted inFig. 7(b) are the wave speeds for case 1. The presence of boundary layer dissipationdoes not qualitatively alter the conclusions of the undamped theory. However, notethat the mean flows impart a splitting of the two sets of waves, one of which isdamped more rapidly than the other. In this case, the upper wave branch (wavesmoving westward more slowly) corresponds to the set of slow modes. We describethe linear theory of case 2 without a figure. In case 2 there is also no thermal dissipationand mean flows have the opposite sign of case 1; a barotropic mean shear of �2.5ms�1

and a baroclinic mean shear of 3.6 ms�1. In this case, though the fast-mode damps atabout 17 days the slow-2mode timescale drops from 244 days to less than 100 days over

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FIGURE 7 Case 1, dd� ¼ 0, positive barotropic wind at the base of the free troposphere at the equator.(a) Damping time as a function of zonal wave number. (b) Wave speed in the stationary frame as a function ofzonal wave number. The upper branch corresponds to the slowly damped waves.

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FIGURE 8 Case 4, dd� ¼ 0:67, negative barotropic wind at the base of the free troposphere at the equator.(a) Damping time as a function of zonal wave number. (b) Wave speed in the stationary frame as a function ofzonal wave number. The lower branch corresponds to the more slowly damped waves.



the first six wavenumbers. Again the wave speeds are not significantly altered, howeverthe slowly damped waves travel westward with greater velocity.

Cases 3 and 4 have dd� ¼ 0:67. In case 3 (not plotted) the mean zonal barotropicshear is 2.5ms�1 making the baroclinic wind �3ms�1. The damping time for the fastand slow modes do not change significantly with wavenumber, the former beingabout 15 days and the latter about 60 days. The wave speeds have the same qualitativebehavior as in the nondissipated case and again the weakly damped waves travelwestward more slowly than the strongly damped waves. Case 4 uses a mean barotropicshear of �2.5ms�1 so that the baroclinic shear is 3ms�1. Figure 8(a) shows that theslow modes do not remain well separated from the fast modes, both having a timescaleof approximately 25 days at wavenumber six. Clearly the designations ‘‘slow’’ and‘‘fast’’ are no longer valid. The wave speeds are qualitatively the same as the earliercases with the slow branch traveling westward with greater velocity.

In conclusion the linear theory of dissipated baroclinic barotropic waves exhibits amyriad of different behaviors. The general features of the wave speeds appear to beunaffected by dissipation. When there is no thermal dissipation, one set of waves israpidly damped while the other remains slowly damped for all of the asymptoticallyvalid wavenumbers. However, for significant amounts of thermal dissipation the twobranches do not remain well separated, both rapidly decaying.

5. NUMERICAL SIMULATIONS OF THE AMPLITUDE EQUATIONS

IN THE PRESENCE OF DISSIPATION

We integrate (53) over a zonal domain which corresponds to the circumference of theEarth. The numerical integrations reported here were performed using a de-aliasedpseudo-spectral method with at least 64 modes of spatial resolution combined with afourth-order Runge–Kutta time discretization. For the simulations describing solitarywaves, we used 256 modes of resolution. Conservation of energy to within 10�6 is satis-fied for all numerical solutions and as in MB, we ensured that less than 4% of theenergy resided in modes higher than k¼ 8 for all times in order to be consistent withthe long wavelength asymptotics. In order to reconstruct the velocity fields the standardvalue of c ¼ 50ms�1 and the choice � ¼ 0:1 is used.

With regard to the plots, the waves in all of the flow vector figures are shown in thecarrier wave frame of reference, �16:6ms�1. Furthermore, the velocity vectors in thetroposphere are plotted for the same example at all times using the same scale; bothof the solitary waves use the same scales. The vectors in the boundary layer arestretched in order to highlight the flow features, but also use the same scales for agiven example.

In this section we consider four examples of dissipative equatorial baroclinic andbarotropic waves in the case m¼ 1. This corresponds to a symmetric baroclinic waveand mean flow interacting with barotropic waves. The antisymmetric barotropicwaves evolve under linear dispersion and dissipation in this situation and, since theyhave no effect on the other waves or mean flows, we always consider zero initialdata for these waves. Therefore the setting is symmetric barotropic and baroclinicmean shears in the presence of symmetric barotropic and equatorial baroclinicwaves. First we consider the effect of boundary layer dissipation on solitary waveswhich arise in the ideal setting using , �� ¼ 2 and �dd� ¼ 0. We then revisit two examples



considered by MB in the ideal context and consider the evolution of the same initialconditions in the presence of both boundary layer and thermal dissipation; �� ¼ 2and �dd� ¼ 0:67. The second example was chosen in order to study the exchange ofenergy from equatorial barotropic Rossby waves with significant midlatitude projectionto equatorial baroclinic Rossby waves which are concentrated around the tropics. Thethird example considers the transfer of energy from the tropical equatorial baroclinicwaves to the barotropic waves with significant midlatitude projection. By comparingthe ideal results in MB with those in the presence of boundary layer drag and thermaldissipation, we can see the effects of dissipation on the rate and amount of energyexchange between the tropical and extra-tropical waves that were discussed in MB.In the final example, we again only use boundary layer dissipation and begin with amean flow which is already in a slaved state. Using random initial wave data we con-sider the long-time evolution of the waves and we shall find that they evolve to a slavedstate of their own with extremely interesting patterns.

5.1. Effect of Boundary Layer Dissipation on Solitary Waves

Biello and Majda (2003) construct solitary wave solutions of the undamped amplitudeequation (53) of the form

Aðx� ctÞ ¼ a sech2x� ct

�

� þ �AA, BSðx� ctÞ ¼ �b sech2

x� ct

�

� þ �BB: ð69Þ

Such exact solutions are archetypes of more general localized structures and provide anidealized setting where the effects mean shear and nonlinear advection could bestudied. We now add the effect of boundary layer dissipation on such localizedwaves. In the setting without dissipation, to every solitary wave there correspondsanother through interchanging A ! �A. This is a reflection of the vertical symmetryof the undamped equations which we expect boundary layer dissipation to break. Thenonzero constants, �AA and �BB give solitary waves in specific baroclinic and barotropicmean shears.

We numerically integrate two solitary wave solutions for 20 days using �� ¼ 2 andd� ¼ 0 and compare the results to the initial waves. The waves have wave amplitudeswhich are comparable to the mean wind and shears. The initial condition of the firstwave is shown in Fig. 9; a 11 400 km wave traveling at �15ms�1 in a barotropicmean wind of 3ms�1 and a baroclinic mean shear of �5ms�1. This soliton in a shearflow has a cyclonic equatorial baroclinic component and an anticyclonic barotropiccomponent. The relation of A and BS is reasonably near that for the slow slavedstate in (62) so we expect boundary layer dissipation to be inefficient here. The windsare much stronger in the top of the troposphere, 8ms�1 (Fig. 9c), compared with�2ms�1 (Fig. 9b) at the base. The boundary layer flow converges at the equatorexcept at the soliton where it is divergent and westerly. Notice that the solitary waveitself has the effect of locally reversing the mean flow in the troposphere. After 20days the wave is shown in Fig. 10. Clearly very little dissipation has taken place, thewave having simply traveled eastward in the moving frame.

The initial condition for second solitary wave in Fig. 11 reverses the sign of thebaroclinic flow component of the previous wave so that both components are anti-cyclones. Here there is a strong projection on the fast component in (62) so we



expect the dissipation to be very efficient. In this instance there is an 8ms�1 mean windat the base of the troposphere and a correspondingly large boundary layer flow whichdiverges from the equator everywhere except in the solitary wave. Again the wind atthe center of the solitary wave is in the opposite direction of the mean wind and attainsa maximum easterly wind speed of 7ms�1 at the base of the troposphere. As a result ofthe vertical baroclinic symmetry, in the absence of dissipation this solution wouldbehave exactly like the previous solution. Figure 12 shows that after only 10 days

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FIGURE 9 Initial condition for a 11 400km solitary wave traveling at �15ms�1. (a) Wave amplitudes,solid – barotropic, dashed – baroclinic. (b) Velocity in the boundary layer. (c) Velocity at the base of the freetroposphere. (d) Velocity at the top of the free troposphere. The flows in all the figures are shown in the carrierwave frame of reference, �16:6ms�1. For all of the solitary waves, the tropospheric flow vectors are plottedwith the same scaling as are the boundary layer flow vectors, the latter being stretched to emphasize flowfeatures.



the solitary wave is barely recognizable having dissipated to about half its strength.Its ghost remains in the persistent boundary layer flow which continues to dissipatethe wave.

We conclude that many of the solitary waves present in the ideal case can be expectedto persist in the presence of boundary layer dissipation. In particular, localized struc-tures which are superimposed on slaved or nearly slaved mean flows and which havethe necessary weak flow at the base of the troposphere can exist quite unaffected bydissipation over intraseasonal timescales. Specifically superimposed cyclonic (anticyclo-nic) barotropic and anticyclonic (cyclonic) baroclinic solitary waves will be long lived

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FIGURE 10 The solitary wave of Fig. 9 after 20 days. (a) Wave amplitudes, solid – barotropic, dashed –baroclinic. (b) Velocity in the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity atthe top of the free troposphere.



even in the presence of boundary layer dissipation. Solitary waves with the same senseof barotropic and baroclinic rotation will tend to dissipate rapidly.

5.2. Effect of Dissipation on Tropical Winds Forced by Midlatitude Waves

MB consider an initial condition with baroclinic wave amplitude equal to zero. Theenergy is concentrated in the barotropic wave train and the baroclinic and barotropicmean flows. There is no barotropic zonal mean shear whereas the baroclinic meanshear velocity is 5ms�1 at the equator. Using the same initial conditions MB discussedhow vertical mean shear arising from baroclinic mean shear provides a route for

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FIGURE 11 Initial condition for another 11 400 km solitary wave traveling at �15ms�1. (a) Wave ampli-tudes, solid – barotropic, dashed – baroclinic. (b) Velocity in the boundary layer. (c) Velocity at the base of thefree troposphere. (d) Velocity at the top of the free troposphere.



midlatitude connections. The initial condition is shown in Fig. 13(a) and the corre-sponding flow field at the bottom and top of the troposphere are shown respectivelyin Fig. 13(c) and (d). The boundary layer flow, which is absent in the ideal case, isshown in Fig. 13(b). It consists of three strong eastward divergent winds at the flowmaxima at the base of the troposphere, which are also the maxima of the barotropicwaves. There are also westward and eastward flows at higher latitudes. These arelocated at the maxima and minima of the barotropic waves and are a reflection ofthe strong winds at midlatitudes associated with the barotropic waves.

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FIGURE 12 The remnants of the solitary wave of Fig. 11 after 10 days. The amplitudes are already greatlydecayed and by 20 days are absolutely negligible. (a) Wave amplitudes, solid – barotropic, dashed – baroclinic.(b) Velocity in the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity at the top ofthe free troposphere.



After 14 days much dissipation has occurred but there has also been a significanttransfer of energy to baroclinic waves. In Fig. 14 strong baroclinic wave activity isevident in the left half of the domain. The dominance of the baroclinic component ofthe velocity field around x ¼ 15 is the remnant of the ‘‘westerly wind burst-like’’event that was described by MB. Furthermore, it is clear from the boundary layerflow, Fig. 14(b), that the dissipation is concentrated around this structure.

As a measure of the effect of dissipation we compare the diagnostics of the ideal caseconsidered by MB at 14 days to those of the dissipative case considered here. In par-ticular the baroclinic mean shear decreases from 5ms�1 in the ideal case to 2:5ms�1

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(a) Amplitudes; Time =0 Days

FIGURE 13 Initial velocity profile of a barotropic wave train with no initial barotropic mean windand baroclinic mean shear of 5ms�1. (a) Wave amplitudes, solid – barotropic, dashed – baroclinic.(b) Velocity in the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity at the topof the free troposphere.



in the presence of dissipation. There is actually barotropic mean shear generation in thisexample with the mean shear decreasing from 0 to �0:75ms�1, consistent with its driveto a slaved mean flow state. For the energy diagnostics we define �D to be the ratio ofenergies at 14 days in the dissipative example to that in the ideal example. D denotes theenergy considered and shall take the values D ¼ tot,A,B which are the total energy,baroclinic wave energy and barotropic wave energy, respectively. The numerical resultsyield

�tot ¼ 0:36, �A ¼ 0:22, �B ¼ 0:62: ð70Þ

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FIGURE 14 Velocity profile 14 days after an initially barotropic wave train with no initial barotropic meanwind and baroclinic mean shear of 5ms�1. The ‘‘westerly wind burst’’-like event persists despite both bound-ary layer and thermal dissipation dissipation. (a) Wave amplitudes, solid – barotropic, dashed – baroclinic.(b) Velocity in the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity at the top ofthe free troposphere.



Clearly the dissipation is effective at inhibiting baroclinic wave forcing by barotropicwaves. However as is evident from Fig. 14, the most interesting strong localized struc-tures do persist. After 14 days, the ratio of baroclinic to barotropic wave energies in theideal case was 1.1 indicating a strong connection whereas in the dissipative setting theconnection is much weaker with the ratio being 0.4.

5.3. Effect of Dissipation on Midlatitude Waves Driven by Tropical Winds

MB considers the transfer of energy from an initial equatorial baroclinic wave train tobarotropic waves with significant midlatitude projection. Using a baroclinic initialmean shear of 5:0ms�1, a barotropic initial mean shear of 2:5ms�1 and the same initialconditions of MB, the equations are numerically integrated in the presence of bothboundary layer and thermal dissipation. The initial condition is shown in Fig. 15(a),the baroclinic wave and mean amplitude being the dashed curve while the barotropicwave and mean amplitude is solid. The tropospheric flow associated with this wavetrain (Fig. 15c and d) is highly antisymmetric between the top and bottom. The flowmainly consists of two localized baroclinic structures which are westerly at theground plus an easterly wind leading located near x¼ 35. The boundary layer flowin Fig. 15(b) is dominated by the two divergent, westerlies associated with the barocliniccyclones in the troposphere. The easterly, convergent boundary layer flow associatedwith the easterly wind at the base of the troposphere is weaker than its westerly counter-parts. The boundary layer flow at higher latitudes is much weaker than the previousexample since the barotropic amplitudes are much smaller here.

Figure 16 shows the amplitudes and flow fields after 14 days. Though the baroclinicwaves are much dissipated, there is yet a significant transfer of energy to the barotropicwaves in the form of the strong barotropic wave packet around x¼ 15. Nonetheless thestrongest boundary layer winds occur near this packet and are divergent westerlies atthe equator with easterly return flows at higher latitudes; indicative of the overlyingbarotropic wave.

In this example, the dissipative dynamics dramatically reduces the mean flows. At14 days the baroclinic mean shear is 1:75ms�1 while the barotropic mean shear is0:71ms�1. The ratio of the dissipated energies to the original energies are

�tot ¼ 0:24, �A ¼ 0:13, �B ¼ 0:29: ð71Þ

The energy of the dissipative waves is greatly reduced even in comparison to theprevious example. As a measure of the efficacy of wave energy exchange, the ratio ofbarotropic to baroclinic wave energy in the nondissipative case is 0.53 whereas in thedissipative case the ratio is 0.59. Since the two ratios are nearly equal it seems that dis-sipation does not greatly affect wave energy transfer in this example.

In summary, it appears that dissipation is very effective at modifying the meanflow of the waves which, in turn, can affect the energy transfer. Nonetheless coherentstructures which exist in the nondissipative examples persist in the dissipativeexamples. The basic qualitative result that there is significant energy exchange betweenthe midlatitudes and the tropics in the presence of baroclinic mean shears is not affectedby the dissipation.



5.4. Dissipation Balanced States

Since the previous examples suggest that the modification of the mean flows is theprimary effect of dissipation on barotropic and baroclinic wave energy exchange, it isuseful to consider initial mean shear profiles which do not change significantly underdissipation. In this section we consider the reduced equations for equatorial baroclinicand barotropic waves only in the presence of boundary layer dissipation; thermaldissipation is set to zero. We select initial conditions so that the mean flows are alreadyslaved according to (62), the baroclinic mean shear is �4:5ms�1 at the base of the

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FIGURE 15 Initial velocity profile of a baroclinic wave train with barotropic mean wind of 2:5ms�1

and baroclinic mean shear of 5ms�1. (a) Wave amplitudes, solid – barotropic, dashed – baroclinic.(b) Velocity in the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity at the topof the free troposphere.



troposphere while the barotropic mean shear is 3:1ms�1 at the base of the troposphere.The barotropic and baroclinic waves are initialized with random wave packets and theinitial conditions, waves and means, are shown in Fig. 17(a). The wind profile showsvery strong barotropic waves near x¼ 5 and 25 while there is a mixed barotropic/baroclinic packet near x¼ 32. The boundary layer flow in Fig. 17(c) has a bit ofeverything. There are convergent easterly and divergent westerly flows near the equatorwhen there are strong winds in the overlying troposphere. Furthermore there are easte-rly flows at higher latitudes over regions where the tropospheric waves are stronglybarotropic.

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FIGURE 16 Velocity profile 14 days after an initially baroclinic wave train with barotropic mean wind of2:5ms�1 and baroclinic mean shear of 5ms�1. A strong barotropic wave packet near x ¼ 20 indicates thatdissipation does not inhibit the conversion of baroclinic wave energy to barotropic wave energy. (a) Waveamplitudes, solid – barotropic, dashed – baroclinic. (b) Velocity in the boundary layer. (c) Velocity at the baseof the free troposphere. (d) Velocity at the top of the free troposphere.



These initial conditions are integrated for long times compared to the other results wehave presented. After 60 days of interaction the waves settle into a slowly decaying state(shown in Fig. 18) which persists for much later times. At this time the zonal meanbaroclinic shear is �3:5ms�1 and the barotropic mean shear is 2:4ms�1, havingchanged little from their initial conditions. The amplitudes in Fig. 18(a) bear a strikingresemblance to a packet of multiple slaved solitary waves of the type shown in Fig. 9.In fact, all of the wave packets travel at almost the same velocity, �11ms�1 with respectto the stationary frame. The barotropic amplitude has a clear plateau level with threelocalized wave packets at x ¼ 11, 20 and 33 which oppose the mean barotropic shear.

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FIGURE 17 Random initial wave packet on a mean baroclinic shear of �4:5ms�1 and a mean barotropicwind of 3:1ms�1, which are in a balanced state. (a) Wave amplitudes, solid – barotropic, dashed – baroclinic.(b) Velocity in the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity at the top ofthe free troposphere.



Similarly the baroclinic amplitudes have a clear plateau of the opposite sign of the baro-tropic plateau with three wave packets at the same location of the barotropic packetsand which also oppose the baroclinic mean shear. The velocity field is extremely weakat the bottom of the troposphere being most dominant in the parts of the domaindescribed as the plateau, in particular around x¼ 5 and 26. The boundary layer velocityis easterly and convergent in the regions where the dissipation is greatest.

At the top of the troposphere in Fig. 18(d) the flow is again reminiscent of Fig. 9(d).The centers of these solitary wave-like packets are anticyclonic vortices at the top of thetroposphere. In the domains of the amplitude plateaus there are strong westerly winds

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FIGURE 18 Sixty days after Fig. 17 the waves become slaved to a slowly decaying state which is reminis-cent of collection of solitary waves. (a) Wave amplitudes, solid – barotropic, dashed – baroclinic. (b) Velocityin the boundary layer. (c) Velocity at the base of the free troposphere. (d) Velocity at the top of the freetroposphere.



at the equator which, when the plateau encounters a wave packet, turn around intolong-cyclonic gyres.

To summarize, random initial wave packets superposed on slaved mean flows havea tendency to settle into a collection of solitary wave-like packets over a slowly decayingmean flow state.

6. CONCLUDING DISCUSSION

We have derived a reduced model for the interaction of equatorial baroclinic and baro-tropic waves in the presence of boundary layer and thermal dissipation. The equationsarise through the same long-wave, low Froude number scaling of MB. The ratio ofmeridional to zonal flows in the troposphere is proportional to the small ratio ofmeridional length scales to zonal length scales. Conversely, the dominant balance inthe atmospheric boundary layer allows meridional flows to be of the same order aszonal flows there. The lowest order asymptotics of the boundary layer are simply theequatorial Ekman equations. Therefore meridional and zonal boundary layer flowsare proportional to the negative of the meridional pressure gradient in the overlyingfree troposphere. Equatorial high pressures at the base of the free troposphere drivedivergent easterly flows in the boundary layer, enhancing momentum dissipation.Similarly, low-pressure centers at the bottom of the troposphere drive convergenteasterly flows in the boundary layer, also a source of dissipation. When coupled tothe equatorial baroclinic and barotropic long-wave equations the lowest order sourceof boundary layer dissipation arises from the meridional component of the boundarylayer flow. Therefore the dynamics drive the boundary layer flow to a minimummeridional velocity. This implies that the dynamics in the troposphere tends tominimize the pressure at the base or equivalently minimize the zonal velocity there.When incorporated into the amplitude equations (DREBB), this tendency is manifestedas a near degeneracy of the boundary layer dissipation matrix. The presence of thermaldissipation breaks this degeneracy.

The results of numerical simulations indicate that boundary layer and thermal dissi-pation can significantly reduce the zonal mean flows unless they are initialized withsmall zonal velocity at the base of the free troposphere. The dissipation of meanflows, in turn, can inhibit wave energy exchange between the tropics and midlatitudes.In general the boundary layer dissipation is very efficient for barotropic/barocliniccyclones or anticyclones but is much weaker for barotropic and baroclinic waves ofopposite rotation. Strong coherent structures which form are less effected by the dissi-pation of mean flows since their energy transfer tends to be through direct wave–waveinteraction, not mediated by baroclinic zonal mean flows. Initial wave trains interactingin the presence of balanced zonal mean climate states themselves evolve to a slavedwave state and persist over seasonal timescales. The dissipative balanced state consistsof a collection of coherent wave packets similar to solitary waves for which the flow atthe base of the free troposphere is minimum.

The systematic inclusion of boundary layer forcing can be applied to other asympto-tic regimes. For example, Majda and Klein (2003) have derived scalings from theplanetary to the synoptic scales in which there are resonant midlatitude to tropicsconnections. They can also be included in a straightforward manner in the more generaltheory equatorial Rossby waves in mean shears derived by Biello and Majda (2003).



Finally, the unforced waves we have considered always decayed with time. However,the atmosphere is subject to seasonally varying thermal and convective forcingwhich, in the presence of damping could force the waves and mean flows into newbalanced states. The inclusion of this seasonal forcing would provide an importantextension of the theory which we shall consider in future work. Another interestingdirection is to utilize these models for the equatorial ocean in a different parameterregime.

Acknowledgments

We would like to thank our referees for a careful reading of the manuscript and theiruseful suggestions. The research of A.J. Majda is partially supported by NSF grant #DMS-9972865, ONR grant # N0014-96-1-0043 and NSF DMS-0139918. The researchof J.A. Biello was supported by an NSF VIGRE Postdoctoral Fellowship, # DMS-0083646 at RPI.

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