Wave radiation

A short course on:

Modeling IO processes and phenomena

INCOISHyderabad, India

November 16−27, 2015

References1) HIGnotes.pdf: beginnings of Sections 3−5.

McCreary, J.P., 1980: Modeling wind-driven ocean circulation. JIMAR 80-0029, HIG 80-3, Univ. of Hawaii, Honolulu, 64 pp.

2) KelvinWaves.pdf: A write-up of the Kelvin-wave solution.

Let q be u, v, or p of the LCS model. To focus on free waves, neglect forcing, friction and damping terms. Then, equations of motion for the 2-d qn(x,y,t) fields are

Solutions to these equations describe how waves associated with a single vertical mode propagate horizontally.

Waves associated with a superposition of vertical modes

propagate both horizontally and vertically.

Mode equations

Solving the unforced, inviscid equations for a single equation in vn, and for convenience dropping subscripts n gives

Problem #1: Solve the equations of motion to obtain (1).

(1)

vn equation

Okay. This equation is so important that maybe we should derive it in class!

Derivation of vn equation

(−1)

Derivation of vn equation

(−1/cn2)

Derivation of vn equation

Solving the unforced, inviscid equations for a single equation in vn, and for convenience dropping subscripts n gives

Solutions to (1) are difficult to find analytically because f is a function of y and the equation includes y derivatives (the term vyyt). There are, however, useful analytic solutions to approximate versions of (1).

(1)

vn equation

The simplest approximation (mid-latitude β-plane approximation) simply “pretends” that f and β are both constant. Then, solutions have the form of plane waves,

Then, we can set ∂t = −iσ, ∂x = ik, and ∂y = iℓ in (1), resulting in the dispersion relation,

The dispersion relation provides a “biography” for a model. It describes everything about the waves it supports.

Dispersion relation of free waves

The simplest dispersion relation has f = 0, in which case the waves are non-dispersive gravity waves.

Gravity waves with f = 0

The phase speed of the waves is σ/k = ±c. The property that dispersion curves are linear (straight lines) indicates that the waves are non-dispersive.

When ℓ ≠ 0, the curves define a surface. At each σ, the disp. rel. gives a circle of radius r = σ/c, so the surface is a circular cone.

σ/f

k/α

-1

−1 1

- -

For convenience, the plot shows curves for ℓ = 0.

α = f/c = R−1

Gravity waves with constant f

σ/f

k/α

-1

−1 1

- -

For convenience, the plot shows curves when ℓ = 0.

When f ≠ 0 and is constant, the possible waves are dispersive, gravity waves. There are no waves with frequencies < f.

The phase speed, σ/k, is no longer linear, indicating that the waves are dispersive.

σ/f

k/α

-1

−1 1

- -

f = 0

When ℓ ≠ 0, the curves define a surface. At each σ, the disp. rel. is a circle with r = (σ2−f2)½/c and its center at k = ℓ = 0. So, the surface is a circular bowl.

Gravity waves with variable f (β ≠ 0)

When f ≠ 0 and β ≠ 0, the waves are still dispersive, gravity waves, but the curves are modified by the β term.

When ℓ ≠ 0, the disp. rel. still defines a circle for each σ with its center at k = −β/(2σ), ℓ = 0 and its radius modified from (σ2−f2)½/c. So, the surface is still a circular bowl.

For convenience, the plot shows curves for ℓ = 0. σ/f

k/α

-

-1

-

−1 1

Rossby waves

When σ is small, the σ2/c2 term is small relative to f2/c2, giving the disp. rel. for RWs.

Rossby exist only for negative k, and so propagate westward. σ/f

k/α

-

-1

-

−1 1

R/2Re

When ℓ ≠ 0, the disp. rel. still defines a circle for each σ with its center at k = −β/(2σ), ℓ = 0 and a radius r = β2/(4σ2) − f2/c2.

Freq. σ attains a maximum value when r → 0, that is, when σ = ½(c/f)(β/f) = ½R/Re. So, the surface is an inverted bowl. Typically, R/Re « 1, so that the RW and GW bands are well separated.

The coastal KW propagates along coasts at speed c with the coast to its right, and decays offshore with the decay scale c/f = R, the Rossby radius of deformation.

Kelvin waves

To derive the dispersion relation for GWs and RWs, we solved for a single equation in v. So, we missed a wave with v = 0, the coastal Kelvin wave.

The dispersion curves shown in the figure and equation are for Kelvin waves along zonal boundaries. KWs also exist along meridional boundaries.

σ/f

k/α

-

-1

-

−1 1

Problem #2: Solve the equations of motion to obtain the Kelvin-wave solutions.

Okay. The solution is easy, insightful, and important, so maybe we should derive it in class!

Derivation of KW solution

(−c2)

Derivation of KW solution

(−1)

Derivation of KW solution

Look for solutions proportional to exp(ikx –iσt). Set ∂t = −iσ and ∂x = ik.

Phase and group speed

σ/f

k/α

-

-1

-−1 1

R/2Re

The figure shows the wave types that we have discussed.

The phase speed of a wave with wavenumber k and frequency σ is the slope of the line that extends from (0,0) to (σ,k).

The group speed of a wave with wavenumber k and frequency σ is the slope of the line parallel to the dispersion curve at the point (σ,k).

Movies A1, A3, A2