Wave radiation A short course on: Modeling IO processes and phenomena INCOIS Hyderabad, India November 16−27, 2015

References 1) HIGnotes.pdf: beginnings of Sections 3−5. McCreary, J.P., 1980: Modeling wind-driven ocean circulation. JIMAR , 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 q n (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 v n, and for convenience dropping subscripts n gives Problem #1: Solve the equations of motion to obtain (1). (1) v n equation Okay. This equation is so important that maybe we should derive it in class!

Derivation of v n equation (−1)

Derivation of v n equation (−1/c n 2 )

Derivation of v n equation

Solving the unforced, inviscid equations for a single equation in v n, 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 v yyt ). There are, however, useful analytic solutions to approximate versions of (1). (1) v n 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/fσ/f k/αk/α - 1 −1 1 For convenience, the plot shows curves for ℓ = 0. α = f/c = R −1

Gravity waves with constant f σ/fσ/f k/αk/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/fσ/f k/αk/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 = (σ 2 −f 2 ) 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 (σ 2 −f 2 ) ½ /c. So, the surface is still a circular bowl. For convenience, the plot shows curves for ℓ = 0. σ/fσ/f k/αk/α - 1 −1 1

Rossby waves When σ is small, the σ 2 /c 2 term is small relative to f 2 /c 2, giving the disp. rel. for RWs. Rossby exist only for negative k, and so propagate westward. σ/fσ/f k/αk/α - 1 −1 1 R/2R e 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 = β 2 /(4σ 2 ) − f 2 /c 2. Freq. σ attains a maximum value when r → 0, that is, when σ = ½(c/f)(β/f) = ½R/R e. So, the surface is an inverted bowl. Typically, R/R e « 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/fσ/f k/αk/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 (−c 2 )

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/fσ/f k/αk/α - 1 −1 1 R/2R e 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