Effects of equatorially-confined shear flow on MRG and Rossby waves

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Graphical Abstract

Eﬀects of equatorially-conﬁned shear ﬂow on MRG and Rossby

waves

Mukesh Singh Raghav, Sharath Jose, Amit Apte, Rama Govindarajan

The structure of equatorial waves is greatly aﬀected by the presence of

shear ﬂow. When compared with the zero mean ﬂow case, these waves are

more (less) equatorially trapped for an easterly (westerly) mean ﬂow. In the

β-plane setting, the meridional extent of the waves is underestimated even

for highly equatorially conﬁned mean ﬂows. Our study thus highlights the

discrepancies in the wave characteristics even under favourable conditions for

comparison between the β-plane and the spherical systems.

arXiv:2210.05962v1 [physics.flu-dyn] 12 Oct 2022

Highlights

Eﬀects of equatorially-conﬁned shear ﬂow on MRG and Rossby

waves

Mukesh Singh Raghav, Sharath Jose, Amit Apte, Rama Govindarajan

•Stability analysis of an equatorially conﬁned shear ﬂow is conducted

using beta plane and full spherical equations.

•The structures of equatorial waves are shown to diﬀer from the classical

waves without shear.

•It is shown that even for shear which is highly equatorially conﬁned,

the resulting atmospheric waves are not.

•The full spherical stability analysis provides qualitatively diﬀerent waves

from the waves in beta plane theory.

Eﬀects of equatorially-conﬁned shear ﬂow on MRG and

Rossby waves

Mukesh Singh Raghava, Sharath Josea, Amit Aptea,b,∗, Rama

Govindarajana

aInternational Centre for Theoretical Sciences, Tata Institute of Fundamental

Research, Bengaluru, 560089, India

bDepartment of Data Science, Indian Institute of Science Education and

Research, Pune, 411008, India

Abstract

Linear modal stability analysis of a mean zonal shear ﬂow is carried out in

the framework of rotating shallow water equations (RSWE), both under the

β-plane approximation and in the full spherical coordinate system. Two base

ﬂows – equatorial easterly (EE) and westerly (EW) – with Gaussian proﬁles

highly conﬁned to small latitudes are analyzed. At low Froude number,

mixed Rossby-gravity (MRG) and Rossby waves are found to be particularly

aﬀected by shear, with prominent changes at higher wavenumbers. These

waves become practically non-dispersive at large wavenumbers in EE. The

perturbations are found to be more conﬁned equatorially in EE than in EW

with the degree of conﬁnement being more pronounced in the β-plane system

compared to the full spherical system. At high Froude number, the phase

speeds are signiﬁcantly larger in the β-plane system for all families of waves.

Under the β-plane approximation, exponentially unstable modes can be ex-

cited, having negative (positive) phase speed in EE (EW). Strikingly, this

ﬂow is always neutrally stable with the full spherical system. This speaks for

the importance of studying the whole spherical system even for equatorially

conﬁned shear.

Keywords: Shallow water equations; β-plane approximation; spherical

coordinates; MRG waves; Rossby waves

∗Corresponding author

Email address: apte@iiserpune.ac.in (Amit Apte)

Preprint submitted to Dynamics of Atmospheres and Oceans October 13, 2022

1. Introduction

The theory of equatorial waves is crucial for an understanding of equato-

rial dynamics in geophysical systems (Kiladis et al., 2009). In a seminal work,

Matsuno (1966) derived analytical solutions to the rotating shallow water

equations (RSWE) on an equatorial β-plane linearised about a steady state

with zero ﬂow. Matsuno’s analysis identiﬁed diﬀerent families of waves – east-

ward inertio-gravity (EIG), westward inertio-gravity (WIG), mixed Rossby

gravity (MRG), Rossby and Kelvin waves. Subsequent observations estab-

lished the presence of these waves in the atmosphere and the ocean (Yanai

and Maruyama, 1966; Lindzen and Holton, 1968; Wallace and Kousky, 1968;

Holton and Lindzen, 1968; Takayabu, 1994).

These equatorial waves are known to have notable eﬀects on the dynamics

of the atmosphere and the ocean (Vallis, 2017; Boyd, 2018). For instance,

the modiﬁcation of the zonally symmetric ﬂow in the stratosphere is inti-

mately linked to the MRG wave (Wallace, 1973). Further, Kelvin waves are

known to generate an upward ﬂux of westerly momentum, accounting for

westerly acceleration associated with the quasi-biennial oscillation (Wallace

and Kousky, 1968). The agreement of the prominent peaks in the global

space-time spectrum of the tropical cloudiness (Wheeler and Kiladis, 1999)

and the rainfall (Cho et al., 2004) with theoretically calculated dispersion

curves has proved to be a remarkable success for the theory.

The Matsuno theory is based on zero mean ﬂow, but meridionally aver-

aged climatological zonal ﬂow shows signiﬁcant variation with latitude and

presence of longitudinal shear. Thus a natural question is to understand how

the equatorial waves are modiﬁed in the presence of such shear ﬂows - this

is one of main aims of the present study. We ﬁrst review some of the main

results of previous studies in this direction.

Beyond the primary eﬀects of Doppler shifting (Yang et al., 2003), mean

ﬂow is shown to have non-trivial eﬀects on the properties of equatorial waves

and vice versa (Dias and Kiladis, 2014; B¨uhler, 2014). Critical layers, where

the phase speed equals the mean ﬂow velocity and disturbance energy is

absorbed, can exist (Bennett and Young, 1971). The longitudinal variation

of the mean ﬂow can result in regions where wave energy can be accumulated

or depleted; here, wave trains extending beyond the equatorial regions can

allow for energy exchange with higher latitudes (Webster and Chang, 1988).

MRG and Rossby modes show a higher degree of trapping in the presence of

equatorial easterly than in a westerly mean ﬂow (Zhang and Webster, 1989).

While the β-plane approximation often yields good insights, it must be

appreciated that the dynamics on the sphere can assume greater complex-

ity (Hough, 1898). In a landmark work, Longuet-Higgins (1968) described

in great detail the numerically computed eigenfunctions on the sphere for

the complete range of (a parameter characterising rotation that we deﬁne

in section 2.1), and further discussed the asymptotic forms of these eigen-

functions in various limits of that revealed interesting properties of gravity,

planetary and Kelvin waves. Recent work (Paldor et al., 2018) showed the

solutions in the weak rotation regime to be suitable for most wavenumbers

while the solutions in the strong rotation regime seem to work well for only

small wavenumbers. The Kelvin mode, which was identiﬁed as a distinct

mode in the β-plane conﬁguration, was found to be the lowest order EIG

mode in the spherical system (Garﬁnkel et al., 2017). In the presence of

strong polar and equatorial jets, Paldor et al. (2021) found the shallow layer

depth to be a crucial factor for the growth rates of modal instabilities.

The following two comprehensive studies of stability of RSWE with a

longitudinal shear ﬂow are most relevant to the present paper: Zhang and

Webster (1989) using the β-plane approximation, and Paldor et al. (2021)

using the full spherical system. Zhang and Webster (1989) studied the case

of a mid-latitude jet in addition to equatorial easterly (EE) or westerly (EW)

jets. These proﬁles have features of a realistic ﬂow, but they extend beyond

the equatorial region. So in principle, such ﬂows should be studied in a com-

plete spherical system rather than under the β-plane approximation. Also,

this early study was restricted to small wavenumbers. As will be apparent

in our study, the eﬀects of shear ﬂow are signiﬁcant for large wavenumbers,

which is one of the main contributions of our work.

Paldor et al. (2021) studied the stability of equatorial jets in the spherical

system. Their motivation was to compare the instabilities obtained from

approximate theories – non-divergent (ND) and quasi-geostrophic (QG) –

to those from the full shallow water equations. They showed that the full

equations consistently predict smaller disturbance growth rates. We note

that their ﬂow speeds were very large, whereas we study weaker shears which

occur more commonly in the atmosphere. We note further that their work

was primarily interested in the growth rates of unstable modes, and did not

examine the spatial structure of the perturbations, which is another major

contribution of the present paper.

With this background in mind, we now review the salient features and

main results of this paper. We study how a mean ﬂow with shear modiﬁes

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GraphicalAbstractEectsofequatorially-connedshearowonMRGandRossbywavesMukeshSinghRaghav,SharathJose,AmitApte,RamaGovindarajanThestructureofequatorialwavesisgreatlyaectedbythepresenceofshearow.Whencomparedwiththezeromeanowcase,thesewavesaremore(less)equatoriallytrappedforaneasterly(westerly)meanow....

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Effects of equatorially-confined shear flow on MRG and Rossby waves

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