HW2: 1: Geostrophic current: example calculation

P2

P3P( r )

A B

0db

400db

500db

Northern Hemisphere

PGF CF

0

�A =

Z 1000

500�Adp

�B =

Z 1000

500�Bdp

400dbar

.

∂v∂t+u ∂v

∂x+ v ∂v

∂y+w ∂v

∂z+ fu+ 1

ρ∂p∂y

= Ax∂2v∂x2

+ Ay∂2v∂y2

+ Az∂2v∂z2

4 Scale analysis:

VT

UVLx

V 2

LyWVH

fU ?? = AxVLx2

AyVLy2

AzVH 2

WH~ ULx

WH~ ULx,W ~ UH

Lx~10−3

10-7 10-6 10-6 10-6 10-5 ?? = 10-5 10-7 10-7

fu+ 1ρ∂p∂y

= Ax∂2v∂x2

Geostrophy does not hold

2. Arctic – North Atlantic exchange

5. a) NE US coast: SLR?

A B

USA 1b) (Critical thinking) Do you expect the subtropical gyre of the North Atlantic spin up (intensifying) or spin down (weakening), and why? (5pts)

1c) (Critical thinking) Recent climate model studies suggest that global warming accelerates the SLR along the US Northeast coast. Do you expect an intensified or weakened Atlantic Meridional Overturning Circulation, and why? (5pts)

ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 14

1. Coastally trapped waves: continental Shelf Waves (continue)

2. Equatorial waves: (i) Equatorial Kelvin wave;(ii) EQ Rossby wave;(iii) EQ IGW;(iv) Mixed Rossby-gravity wave – Yanai wave

Learning objectives: understand the characters (e.g. energy dispersion, solution, excitation) of the

following waves

Figure: satellite Observed Sea Surface Height anomalies (SSHA) over the Bay of Bengal of the Indian Ocean: coastal Kelvin waves propagate from the eastern equatorial Indian Ocean to the Bay of Bengal (Rao et al. 2010, DSR)

Coastal Kelvin waves: propagate with the coast to its right (left) in Northern (Southern) hemisphere.Solutions are trapped to the coast, decaying away from it exponentially, with an e-folding scale of the Rossby radius of

deformation.

Previous class:

1. Coastally trapped waves: Continental shelf waves

y

Coast

shallow

deep

x

z

y

H(y)Deep water

Shelf

H is depth of water column. Analytic solution, more complicated.

acts as

Topographic Rossby waves

Shelf waves: dispersive.

Propagation: like coastal Kelvin waves;

Mechanism: like Rossby waves: Potential vorticityconservation:

Here, H varies with y because of the shelf;Near the coast, scale is small,

NH, North Coast

2. Equatorial waves(i) The equatorial Kevin wave

a) Dispersion relation:

barotropic or baroclinic mode speed;

Barotropic mode:

First baroclinic mode:

Phase speed and group velocity of Kelvin waves:

Non-dispersive. Both phase and energy propagate eastward. Exist for all frequencies

€

ω

b) Solution:

E-folding scale:

Equatorially-trapped: due to c) Symmetric about the equator (u and p/sea level)d) Forcing. Changing winds with time – symmetric

about the EQ.

EQ y=0

EQ y>0 N.H.

EQ y<0 S.H.

Equatorial Kelvin wave structure

Red contours: Symmetric about the equator for Sea surface height (SSH)

Satellite-observed sea surface height anomalies

Propagation: direction? Eastward

Westerly wind burst (WWB) associated with the Madden-Julian Oscillation (MJO):

T=t1 T=t2 EastEQWest

EQ

Equatorial (EQ) westerly wind burst

WWB, EQ Kelvin wave &onset of the 1997 El Nino

3. Equatorial Rossby wavesa) Dispersion relation

Here, 𝑙=1,2,3,….is the order number of Hermite function in y direction. Often called “meriodinal mode” number.

Long Rossby waves:

𝜔 = −𝛽𝑘

𝑘'()*(',(-)

𝜔=- *',(-

𝑘As before, c is either a baroclinic or barotropic mode speed.

Long Rossby waves: non-dispersive.Propagates westward. The fastest speed for a baroclinic mode is for 𝑙=1 (first meridional mode Rossby wave):

Which is 1/3 of Kelvin wave speed. So, Rossby waves propagate slower than Kelvin waves

𝑐0 = 𝑐1 = − *',(-

𝑐0 = 𝑐1 = − *'(-

= − *2.

Short Rossby waves:

Phase propagates westward, energy eastward.