Lecture 6: Quasigeostrophic Rossby Waves 6.1 Potential Vorticity Equation

Lecture 6: Quasigeostrophic Rossby waves 6.1 Potential vorticity equation

In Lecture 3, we derived the properties of Rossby waves in the reduced- Consider small-amplitude disturbances to a uniform zonal background gravity model, linearised about a state of rest. Our key ﬁndings were: ﬂow (U, 0, 0) for which the streamfunction is

Ψ = −Uy • Rossby waves rely on the β-eﬀect for their existence; and the quasigeostrophic potential vorticity is • their zonal phase propagation is westward; Q = βy. • their zonal group propagation is westward for long waves and eastward for short waves. The streamfunction for the total ﬂow (background plus small disturbance) is thus We also discussed the mechanism for Rossby wave propagation through ψ = −Uy + ψ", potential vorticity conservation. and the potential vorticity

Since Rossby waves occur at low Rossby number, it is reasonable to try q = βy + q" and model them using the quasigeostrophic equations. where ∂ f 2 ∂ψ" q" = ∇2ψ" + 0 . h ∂z N 2 ∂z    We now substitute the above in the quasigeostrophic potential vorticity equation (5.15) and neglect terms that are quadratic in the disturbance quantities: ∂ ∂ ∂Q + U q" + v" =0 %∂t ∂x& ∂y

2 " " ∂ ∂ 2 " ∂ f0 ∂ψ ∂ψ ⇒ + U ∇hψ + + β =0. (6.1) % &   2  ∂t ∂x  ∂z N ∂z  ∂x     The first term represents the rate of change of the perturbed potential vorticity following the mean flow, and the second term represents the Carl-Gustaf Rossby (1898-1957) advection of mean potential vorticity by the perturbed flow.

FGDF 6-1 FGDF 6-2 6.2 Dispersion relation 6.3 Vertical propagation

We now look for plane-wave solutions, For given real values of k and l (e.g., imposed by land-sea contrasts or orography), we obtain vertical propagation when m is real and nonzero. ψ" = Aei(kx+ly+mz−ωt). Vertical propagation therefore corresponds to m2 > 0, i.e., We shall also assume constant stratiﬁcation, i.e., N 2 = constant. (x) β U − cp = 2 2

0 < U < Uc. (6.4) the wave crests and troughs move westward relative to the background

flow, although they can move eastward relative to the ground for a suf- Stationary waves propagate vertically only in eastward background flows ficiently strong background flow. The latter explains why mid-latitude (U>0) that are not too strong (U < Uc). atmospheric weather systems generally propagate eastward. Moreover, since Uc increases with increasing horizontal wavelength, long waves propagate vertically under a wider range of eastward flows than short waves.

FGDF 6-3 FGDF 6-4 This is consistent with observations: (θ on p = 50 mb, McLandress 2005) Rossby waves are strongly dispersive: an initial disturbance composed of a number of wavelengths will tend to break up, or disperse, as the different • In the winter stratosphere, winds are eastward and stationary Rossby wavelength components propagate away at different phase speeds. waves have large horizontal scales. We can define the group velocity:

(x) (y) (z) ∂ω ∂ω ∂ω cg =(cg ,cg ,cg )= , , . % ∂k ∂l ∂m&

For stationary waves, we put cp = ω =0after diﬀerentiation — note that stationary waves can still propagate information, even though their phase surfaces do not move; this is another consequence of their dispersive nature.

In particular, 2 (z) ∂ω 2f0 βkm cg = = 2 2 2 . ∂m 2 2 2 f0 m N k + l + N 2 - . • In the summer stratosphere, winds are westward and stationary Rossby Since k, m>0, we ﬁnd c(z) > 0, i.e., the waves propagate information waves are absent. g upwards.

For upward propagating Rossby waves, the phase surfaces,

kx + ly + mz − ωt = constant,

slope westward with height; this is also observed for stratospheric Rossby waves.

FGDF 6-5 FGDF 6-6 6.4 Vertically-trapped Rossby waves 2 2 βk f0 d A Returning to the dispersion relation (6.2), if k2 + l2 > β/(U − c(x)) then − k2 − l2 A + =0. p %Uk − ω & N 2 dz2 m is imaginary. This separable solution only works if both If we set m = iµ, where µ is real and positive, then the wave solutions are βk d2A N 2 − 2 − 2 2 − 2 k l = λ and 2 = λ 2 A of the form %Uk − ω & dz f0 " i(kx+ly−ωt) −µz ψ = Ae e , where λ is a separation constant. i.e., the waves propagate horizontally but not vertically, and decay with Here we shall consider only waves in which the surface and bottom density height. anomalies vanish, giving the boundary conditions: ∂ψ" dA Equivalently, this happens if =0 ⇒ = 0 (z =0, −H) ∂z dz − (x) − (x) U cp < 0 or U cp >Uc. where z = 0 is the sea surface and z = −H is the sea ﬂoor.

(x) As in Lecture 4, the vertical modes satisfy For stationary waves (cp = 0), vertical trapping occurs if the background λN winds are westward (U<0) or strongly eastward (U > Uc). A = A0 cos z , % f0 & λNH where = nπ n = 0, 1, 2, ... 6.5 Vertical modes f0

The corresponding dispersion relation for the waves is In the ocean, Rossby waves are strongly constrained by the surface and βk bottom boundaries. As in Lecture 4 for internal gravity waves, we can ω = Uk − 2 2 2 . (6.5) 2 2 n π f0 k + l + 2 2 seek Rossby wave solutions with modal structures in the vertical. N H This is equivalent to the dispersion relation for Rossby waves in the Thus we seek solutions of the form reduced-gravity model (3.6) if we set ψ"(x, y, z, t)=A(z)ei(kx+ly−ωt). (n) NH Ld = , (6.6) nπf0 Substituting the above into the potential vorticity equation (6.1) gives known as the Rossby deformation radius for the nth mode.

FGDF 6-7 FGDF 6-8 The deformation radius is inﬁnite for the barotropic mode (n = 0), and increasingly smaller for each baroclinic mode (n ≥ 1).

The ﬁrst baroclinic deformation radius is approximately 1000 km in the atmosphere and 30 km in the ocean.

Illustration of the diﬀerent deformation radii in the atmosphere and ocean as revealed through (nonlinear, breaking) Rossby waves.

FGDF 6-9