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ATOC 5051 INTRODUCTION TO PHYSICAL Lecture 13 Learning objectives: understand causes for midlatitude Rossby waves & coastally-trapped waves, their propagation & energy

Mid- Rossby waves ( β effect) Announcement: Thursday, Oct 15, 4-5pm: office hour go over HW2 & HW3; Apple pencil problem Clarification: Lecture notes & notes by chapters (required materials); Recommended textbooks – do own check on table of content, maybe different definitions (as in research articles) I. waves: Why study them? Scientific interest – most evident phenomena & oceanic adjustment

Societal importance: Gravity waves - coastal inundation, erosion, etc. Rossby waves – ocean circulation & climate II. What do we want to know about waves?

1961 Chile : eastern Pacific

2011 Japan Tsunami Why do Tsunami waves propagate in all directions? Why are they so fast?

How can we learn these? : relationship between frequency & , telling us wave direction, speed & their frequency dependence III. Waves section in this course – how do we study them? a) Ocean waves in non-rotational system f=0 (gravity waves) Use equations of motion & continuity equation to derived dispersion relation; in detail mathematically – why? understand where it is from b) Ocean waves in rotational system f=constant (Inertial gravity waves) c) Ocean waves in rotational system , Rossby waves Open, unbounded ocean d) Effects of boundary: Coastally trapped waves e) Equatorial waves https://oceanservice.noaa.gov/facts/wavesinocean.html Annual period in eastern tropical Pacfic

Comparison of annual cycle anomalies of observed 20 °C depth (left panels) and the Rossby wave model solution (right panels), for four average seasons (indicated to the left of each row). The common color key is at right, with contour interval of 5 m. Positive values (red) indicate deep anomalies and negative values (blue) indicate shallow anomalies. Kessler 1990. JGR Waves in an ocean with a varying f: Rossby waves

In real ocean, varies with latitude Mid-latitude Plane approximation y ∂f 2Ωcosφ β = = z ∂y R Equations of motion Same shallow water equation (as for the f=constant case), except that here,

(i) (ii)

(iii) (Ro<<1, E<<1) (v) Background state:

Depth of ocean Exercise: Write a single equation in v alone & assume periodic waves, we obtain: Dispersion relation:

where c = ± gH To simplify the case, let look at 1-dimensional situation in x direction. We have (i) For high frequency waves with large ω , the dispersion relation can be approximated by:

€

ω

or, € f

0 This is long under the influence of f we obtained in the previous class. It is also called inertial gravity wave (IGW) They are not influenced much by beta-variation of f! (ii) For low frequency waves with small

Then

These waves do not exist in f=constant case; their existence is due to the introduction of They are called Rossby (or planetary) waves. To further understand the wave property, we simplify the above equation.

Since is small for Rossby waves because of their low frequency, ����� (1 + �) ==(1+ �)/≈ 1 + �, for �<<1 Choose ‘-’ sign:

or,

Since is independent of frequency and wavenumber, they are non-dispersive. They are long Rossby waves and propagate westward; speed decreases as latitude increases. Here, c is vertical mode speed: either a baroclinic or the barotropic mode gravity wave speed. Choose the ‘+’ sign,

They are short Rossby waves. propagates eastward but phase propagates westward. They are dispersive. Dispersion curves for free waves in mid-latitude plane:

Short Rossby Long Rossby

Short Rossby waves are hardly seen in the ocean interior because (1) they are too short to be effectively excited by large-scale , (2) mixing in the ocean acts strongly on short and slow waves. Midlatitude IGW �

! T = ≈ 5���� max " TIGW ≤ Tmax

! T = ≈ 30���� min " � TRW ≥Tmax Rossby waves Summary: Rossby waves in mid-latitude plane (i) Existence of Rossby waves: , the variation of Planetary ( ) with latitude ( ),

(ii) They are low frequency waves (with periods of months to decades);

(iii) Long Rossby waves: non-dispersive & propagate westward Relative vorticity: Long-Rossby waves propagate westward: Explanation using potential - + vorticity conservation

Assume flat ocean bottom: H ~ constant Observed mid-latitude Rossby waves by TOPEX altimetry: surface height anomaly (SSHA)

Mid-latitude Pacific 1) Direction of propagation; 2) How does the speed change with the increase of ? Why?

Breakout session Observed mid-latitude Rossby waves by TOPEX satellite altimetry Mid-latitude Pacific Propagate westward; speed decreases as latitude increases; Oceanic adjustment with z a) Initial perturbation

0

b) Gravity wave radiation 0

c) Geostrophy

0 d) Rossby waves B

e) Rossby waves C

f) Equilibrium state Quiescent If there is persistent forcing: the equilibrium state is “Sverdrup-balance”. Will be introduced later. Summary: for Mid-latitude beta plane approximation (i) For high frequency waves Inertial gravity waves (IGW)

IGW

(ii) For low frequency waves Long Rossby waves: Short Rossby Long Rossby Short Rossby waves: