ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY Lecture 15 Learning objectives: understand the characters (e.g. energy dispersion, solution, forcing) of the equatorial waves Equatorial waves: (i) Equatorial Kelvin wave; (ii) EQ Rossby wave (iii) EQ IGW; (iv) Mixed Rossby-gravity wave – Yanai wave Previous class: (i) Equatorial Kelvin wave

Forcing: symmetric wind about EQ

� a) Dispersion relation b) Solution EQ Kelvin

Non-dispersive; Propagate eastward Exist at all frequency Symmetric wave (ii). Equatorial Rossby waves First: discuss dispersion relation; Then: solution a) Dispersion relation �� � = − � �! + (2� + 1) � Here, �=1,2,3,….is the order number of Hermite function in y direction. Often called “meridional mode” number. Long Rossby waves, k~small: cr ! �=- � "#$% As before, c is either a baroclinic or barotropic gravity wave speed. ! # �=- � � = � = − "#$% ! " $%&' Long Rossby waves: non-dispersive. Propagates westward. The fastest speed for a baroclinic mode is for �=1 (first meridional mode Rossby wave): # # � = � = − = − . ! " $&' ( Which is 1/3 of Kelvin wave speed. So, Rossby waves propagate slower than Kelvin waves Short Rossby waves, k is large: �� � = − � �! + (2� + 1) �

�

Phase propagates westward, energy eastward.

Short Rossby EQ Rossby wave: dispersive Long Rossby waves: ! �=- � "#$% Summary: Dispersion relations: equatorial Kelvin & Rossby waves �

Short EQ Kelvin Rossby EQ Rossby Non-dispersive wave: dispersive Long Rossby waves: ! �=- � "#$% # � = � = − Non-dispersive ! " $%&' (ii) Equatorial Rossby waves: Solutions

!"# � = � � cos �� − �� = 2/� � � #$ cos(�� − ��) Parabolic cylinder function of order � Hermite function of order � #! #! %&' $ %(' $ � = 2��[ + ] sin(�� − ��)

$% $% % & "% & #$ !"# & !'# & � = [ − ] sin(�� − ��) " ! '$() '$*) � = 1,2,3, … ������ ���� ���������� mode number

Some uses this as Equatorial Rossby radius of deformation. RW Symmetric Property: can be both symmetric and anti-symmetric Odd � such as � =1: symmetric RW Even � such as �=2: anti-symmetric RW D0 – black; D1 –red; D2-yellow; D3:green; D4:blue; D5: purple

y Southern Northern Hemisphere EQ Hemisphere Forcing: symmetric and anti-symmetric winds. Anti-symmetric winds (such as “meridional winds”) excite anti-symmetric Rossby waves;

Symmetric winds (such as zonal winds symmetric about the equator) causes symmetric Rossby waves;

Frequency: unlike KW, RWs are low frequency waves with periods longer than 1 month. # � = � = − ! " $%&' Odd � such as � =1: symmetric RW Even � such as �=2: anti-symmetric RW Do you see Rossby wave structure in this SSHA figure?

If yes, which part? EQ KW & RW: Eastern and western boundary reflections Incoming Rossby wave Incoming Kelvin Wave Coastal Reflected Kelvin Kelvin Reflected Rossby wave Reflected Short Rossby

West East Easterly wind Off equatorial influence: Western boundary

Coastal Kelvin

EQ Kelvin EQ

Coastal Kelvin (iii). Equatorially-trapped inertial gravity waves (IGW) a) Dispersion relation: �$ = �$�$ + �� 2� + 1 , � =1, 2, … is the order number for Hermite function.

They are similar to the IGWs in mid-latitude, which are gravity waves under the influence of at the equator and of in mid-latitude. The equatorially-trapped IGWs are dispersive; energy and phase can propagate both eastward and westward. � IGW

� = ��(2� + 1) Rossby Kelvin short long Frequency: High frequency waves (usually T: hours, days…) b) Solution. IGWs have the same solution form as the equatorial Rossby waves: they are oscillating in y direction and meanwhile decay poleward. Their e-folding decay scale is equatorial Rossby radius: c) Symmetric property. Can be both symmetric or anti-symmetric about the equator. [Note: if u & p are symmetric, we call them symmetric waves.] d) Forcing. High-frequency gusty winds, tides, etc. (iv). Yanai waves or mixed Rossby-gravity waves (MRGW) a) Dispersion relation b) Solution (it is the � = 0 case) c) Symmetric property: Anti-symmetric d) Forcing: Anti-symmetric winds (say, meridional winds) Yanai waves: Which direction

is Cp? Which direction is Cg? IGWs: high frequency; Tmax=days IGW MRGW (Yanai)

Rossby Kelvin short

long

Rossby waves: low frequency; Tmin=1 month Dispersion relationship for modes of the linear shallow water equations, from Cane and Sarachik (1976). The dashed box near the origin is the region of frequency- wavenumber space relevant to ENSO. Here, n is the meridional mode number, which is our l discussed above and is more common in physical oceanography research So far, we have learnt: Gravity waves (surface & internal) Below: long surface gravity waves: Oceanic bore France Surface gravity waves: long & short

Internal wave (photo from space) Mid-latitude Rossby waves Equatorial waves: Rossby and Kelvin:

https://www.youtube.com/ watch?v=VNefCmc3_1Y Coastal Kelvin wave