Atmospheric Waves Chapter 6 Paul A. Ullrich [email protected] Part 5: Mixed Rossby-Gravity Waves Trapped Waves Trapped waves decay exponentially rapidly away from a boundary (such as the equator or a coastline). Trapped waves include: • Equatorial (and coastal) Kelvin waves • Equatorial Rossby (ER) waves • Mixed Rossby-Gravity (MGR) waves at the equator • Equatorial inertia-gravity waves Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves • Shallow water equations, no topography • Coriolis parameter at the equator is approximated by equatorial β-plane: 2⌦ f βy y ⇡ ⇡ a ✓ ◆ Du @Φ βyv + =0 Dt − @x Dv @Φ + βyu + =0 Dt @y DΦ @u @v + Φ + =0 with Φ = gh Dt @x @y ✓ ◆ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Linearize shallow water equations about a u = u ,v = v ,h= H + h state at rest with mean height H 0 0 0 @u0 @Φ0 βyv0 + = 0 (1) @t − @x @v0 @Φ0 + βyu0 + = 0 (2) @t @y @Φ0 @u0 @v0 + gH + = 0 (3) with Φ0 = gh0 @t @x @y ✓ ◆ Seek wave solutions of the form (allow amplitudes to vary in y): (u0,v0, Φ0)= uˆ(y), vˆ(y), Φˆ(y) exp(i(kx ⌫t)) − ⇣ ⌘ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Yields the system i⌫uˆ βyvˆ + ikΦˆ = 0 (1) − − @Φˆ i⌫vˆ + βyuˆ + = 0 (2) − @y @vˆ i⌫Φˆ + gH ikuˆ + = 0 (3) − @y ✓ ◆ βy k Solve (1) for uˆ : uˆ = i vˆ + Φˆ (4) ⌫ ⌫ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Plug (4) into (2) and (3), rearrange terms @Φˆ β2y2 ⌫2 vˆ ikβyΦˆ i⌫ = 0 (5) − − − @y @vˆ k ⌫2 gHk2 Φˆ + i⌫gH βyvˆ = 0 (6) − @y − ⌫ ✓ ◆ Solve (6) for Φ ˆ : i⌫gH @vˆ k Φˆ = βyvˆ (7) −(⌫2 gHk2) @y − ⌫ − ✓ ◆ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Plug (7) into (5) and rearrange terms. Yields second- order differential equation for amplitude function @2vˆ ⌫2 k β2y2 + k2 β vˆ = 0 (8) @y2 gH − − ⌫ − gH ✓ ◆ More complicated equation than before since the coefficient in square brackets is not constant (depends on y) Before discussing the solution in detail, let’s frst look at two asymptotic limits when either H→ ∞ or β=0 Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Asymptotic limit H→∞ Means that the motion is non-divergent, consequence of the continuity equation (3): @u0 @v0 1 @Φ0 + = 0 for H @x @y −gH @t ! !1 ✓ ◆ Equation (8) becomes @2vˆ k + k2 β vˆ = 0 (9) @y2 − − ⌫ Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Asymptotic limit H→∞ Asymptotic limit H→∞ @2vˆ k + k2 β vˆ = 0 (9) @y2 − − ⌫ Solutions for the amplitude function vˆ in (9) exists of the form: vˆ = v0 exp(i`y) Provided that the frequency ν satisfes the Rossby wave dispersion relationship: βk ⌫ = 2− 2 k + ` It shows that for non-divergent barotropic fow, equatorial dynamics are in no way special Earth’s rotation enters in form of β, not f: Rossby wave Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Asymptotic limit β=0 All infuence of rotation is eliminated. Due to linearity, solutions of (9) exists of the form vˆ = v0 exp(i`y) Equation (8) reduces to the shallow water gravity model @2vˆ ⌫2 + k2 vˆ = 0 (10) @y2 gH − Non-trivial solutions if frequency ν satisfes ⌫ = gH(k2 + `2) ± Pure gravity wave responsep (eastward and westward) Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves In the general case with 2 2 2 2 @ vˆ ⌫ 2 k β y + k β vˆ = 0 (8) @y2 gH − − ⌫ − gH ✓ ◆ we seek solutions for the meridional distribution of vˆ with boundary condition vˆ 0 for y ! | | !1 For small y the coefficient in square brackets is positive and solutions oscillate in y. For large y the coefficient is negative, solutions either grow or decay. Only the decaying solution satisfes boundary condition. Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves Matsuno (1966) showed that decaying solutions only exist if the constant part of the coefficient in square brackets satisfes the relationship pgH ⌫2 k k2 β =2n +1 n =0, 1, 2,... β gH − − ⌫ ✓ ◆ This is a cubic dispersion relation for frequency ν. It determines the allowed frequencies ν of equatorially trapped waves for the zonal wavenumber k and meridional mode number n. Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves In (8) replace y by the nondimensional meridional coordinate β 1/2 ⇠ = y pgH "✓ ◆ # ⇠2 Then the solution has the form vˆ(⇠)=v H (⇠)exp 0 n − 2 ✓ ◆ th Where v0 is a constant and H n ( ⇠ ) designates the n Hermite polynomial. The frst few Hermite polynomials have the form 2 H0(⇠)=1,H1(⇠)=2⇠,H2(⇠)=4⇠ 2 − Index n corresponds to the number of N-S roots in v. Paul Ullrich Atmospheric Waves March 2014 Hermite Polynomials Meridional (N-S) structure of the amplitude function: ⇠2 vˆ(⇠)=v H (⇠)exp n =0, 1, 2,... 0 n − 2 ✓ ◆ n n 2 d 2 Hermite polynomials: H (⇠)=( 1) exp(⇠ ) exp ⇠− n − d⇠n N vˆ(⇠) 0 ξ vˆ(λ) S n = 0 n = 1 n = 2 n = 5 n = 10 Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves In general, the three ν solutions of pgH ⌫2 k k2 β =2n +1 n =0, 1, 2,... β gH − − ⌫ ✓ ◆ can be interpreted as an equatorially trapped 1) Eastward moving inertia-gravity wave 2) Westward moving inertia-gravity wave 3) Westward moving equatorial Rossby wave However, the case n = 0 must be treated separately: For n = 0 the meridional velocity perturbation has a Gaussian distribution centered at the equator Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves For n = 0 (special case) the dispersion relation factors as ⌫ β ⌫ k + k =0 pgH − − ⌫ pgH ✓ ◆✓ ◆ The root ⌫ = k gH (westward propagating gravity wave) is not permitted, violates− an earlier assumptions when eliminating Φˆ p The two other roots are 1 4β ⌫ = k gH 1+ "2 ± s k2pgH # p Paul Ullrich Atmospheric Waves March 2014 Equatorial Rossby and Mixed Rossby-Gravity Waves The positive root corresponds to an eastward moving equatorial inertia-gravity wave The negative root corresponds to a westward moving wave, which resembles • An equatorial (westward moving) inertia-gravity (IG) wave for long zonal scale (k → 0) • An equatorial Rossby (ER) wave for zonal scales characteristic of synoptic-scale disturbances This mode (n=0, westward moving) is therefore called Mixed Rossby-Gravity (MRG) wave. Paul Ullrich Atmospheric Waves March 2014 Equatorial Mixed Rossby-Gravity Waves Plane view of horizontal velocity and height perturbations in the (n=0) Mixed Rossby-Gravity (MRG) wave, propagates westward H L Equator L H Paul Ullrich Atmospheric Waves March 2014 .