Fundamentals of Atmospheric Dynamics focused on Rossby waves Professor In-Sik Kang Content 1. Systematic approximation of governing equations 2. Rossby wave dispersion, wave selection, and barotropic instability 3. Meridional dispersion (propagation) of Rossby waves and teleconnection dynamics 4. Forced Rossby waves by topography 5. Multiple equilibrium
1. Systematic approximation of Governing equations � Reading Materials � Holton (Chapter 1, 2) for the exact equation set and primitive equation, and Geophysical Fluid Dynamics (Pedlosky, Chapter 6) for hydrostatic approximation Before going into the lecture, the governing equation sets of the atmosphere with various complexity, from the most complete set of equations to the simplest possible equation, are introduced in this section. In particular, introduced is how the nondivergent barotropic vorticity equation, which is a simple equation and will be used for most of the topics in this lecture, is obtained from the exact equation set of fluid dynamics. 1) The exact equation set (compressible, inhomogeneous, inertial system) The equation set of fluid dynamics includes (1) the Newton’s second law of motion field, (2) the first law of thermodynamics, (3) the mass conservation, and (4) the ideal gas law. � � D v v � � 1 ∂ (1) ( v ) v p = + ⋅ ∇ = − ∇ − ∇ Φ Dt t ∂ ρ dT dp • (2) c p Q − α = dt dt � D ρ (3) v = − ρ ∇ ⋅ Dt p RT (4) = ρ � The above 4 equation set has four variables ( ), therefore it is a closed system. v , p , , T ρ The exact equation set governs the motion field and thermodynamic property of all kinds of fluids. This equation set contains all kinds of waves, such as the sound waves, 2
gravity waves, and large-scale Rossby waves when the governing equations of motion fields include the Coriolis force, particularly for large scale fluids whose time scale is about or longer than a day (earth rotational time scale). Note that the vorticity equation of the above system is � D � � � � 1 ω . (5) ( ) v ( v ) ( ) p = ω ⋅ ∇ − ω ∇ ⋅ − ∇ × ∇ Dt ρ 2) The primitive equation set 1 p ⎛ ∂ ⎞ => pressure coordinate Assumption: L >> D ⇒ Hydrostatic g ⎜ = − ⎟ ⎜ ⎟ z ρ ∂ ⎝ ⎠ p δ (p & z are in opposite directions.) => pressure gradient g z ρ = δ = δφ Du d φ (6) fv F = − + + u Dt dx Dv d φ (7) fu F = − − + v Dt dy dp => d φ = − (8) g , α = − ρ dz dp dT dp + & (9) c Q = − α p dt dt du dv d dp ω (10) 0, + + = ω = dx dy dp dt (11) p RT = ρ The above primitive equation set is based on the hydrostatic approximation, where the vertical momentum equation becomes a diagnostic equation expressing the 3
balance between the gravity force and the vertical pressure gradient. This system is applicable to the large-scale fluid motion whose horizontal scale is much larger than the vertical scale and therefore the Coriolis forcing terms should be included. This system provides a very good approximation of atmospheric state for a spatial scale of more than few tens of km. Therefore, the regional models as well as global general circulation models usually use this primitive equation set for the weather and climate simulations. 3) Shallow water system (hydrostatic & homogeneous ) ρ = ρ 0 p ∂ , . Hydrostatic equation . Assumption: L >> D g ρ = ρ = − ρ 0 0 z ∂ Integrating the hydrostatic equation from the top of water surface to the height z, p Z , we get . p g z p p g ( H z ) ∫ δ = ∫ − ρ δ = + ρ − 0 0 0 p H 0 p ∇ Now the horizontal pressure gradient is independent of z , which H g H = ∇ H ρ 0 implies that u & v are independent of z , giving the barotropic (vertically uniform) condition. Also the constant density implies that the temperature is proportional to the pressure, which depends on the height H in this system,. Now the governing equations of the shallow water system are du H ∂ (12) g fv = − + dt x ∂ dv H ∂ (13) g fu = − − dt y ∂ 4
u v w ∂ ∂ ∂ 0 + + = x y z ∂ ∂ ∂ � W H H From the above continuity equation, which gives w v z ∫ δ = ∫ − ∇ ⋅ δ H H 0 0 � DH , and since , w ( v ) H w = − ∇ ⋅ = ( z H ) H H ( z H ) = = Dt DH � (14) ( v ) H = − ∇ ⋅ H H Dt Equations (12), (13), and (14) consist of the equation set for the shallow water system. Applying the curl to Eqs. (13) and (14), r d g (15) ( f ) ( f )( v ) dt ζ + = − ζ + ∇ H H Combining Eqs. (14) and (15), we can obtain the following potential vorticity equation. D ς f + ( ) 0 (16) = Dt H Now, for H=constant, the rigid top condition is equivalent to the (horizontal) r u v ∂ ∂ divergence to be zero from Eq. (14), . Then, v 0 ∇ ⋅ = + = H H x y ∂ ∂ D ς (17) ( + f ) 0 = Dt The above equation is called the “nondivergent barotropic vorticity equation,” which will be a base equation of this lecture. Comparing of (17) with (5), we see the simplicity of (17), which contains only the large-scale Rossby waves. The nondivergence condition introduces the streamfunction ( ψ ) as below u v u v ∂ ∂ ∂ ψ ∂ ψ ∂ ∂ => 2 0 u and v + = => = − = ζ = − + = ∇ ψ x y y x y x ∂ ∂ ∂ ∂ ∂ ∂ 5
f ∂ ζ ∂ ζ ∂ ζ ∂ Eq. (17) can be linearized with respect to the basic state u v v 0 + + + = t x y y ∂ ∂ ∂ ∂ of u and , which are assumed to be constant in space and time, and df / dy β = . After neglecting the small nonlinear perturbation terms, v = 0 ' ' ∂ ς ∂ ς u v ' 0 + + β = t x ∂ ∂ The above equation is a simplest possible equation of describing large-scale atmospheric and ocean circulations. It expresses that the vorticity change can be due to the mean advection of relative vorticity and the meridional advection of planetary vorticity. Note that for the planetary-scale waves, the planetary vorticity advection (the third term) is much larger than the relative vorticity advection (the second term). - -------------------------------------------------------------------------------------------------------------- It is also noted that Eq. (15) can be written in a quasi-geostrophic approximation, where f and the absolute magnitude of ζ is much larger than >> ζ that of divergence D , Then, the vorticity equation in a quasi-geostrophic approximation can be written as r ∂ ζ (18) v ( f ) fD + ⋅∇ ζ + = − t ψ ∂ r where v ψ is the streamfunction (non-divergent) component of wind and . It is noted that the wind associated with the advection term in (18) is gH / f ψ = o non-divergent, and therefore the Eq. (18) is same as Eq. (16) except the right-hand side term fD which is often considered as a forcing term of circulation (streamfunction). 6
2. Rossby waves 1) Rossby wave dispersion in a β-plane and wave selection The linearized nondivergent barotropic vorticity equation, driven in the previous section, will a base function in this section. ' ' ∂ ς ∂ ς u v ' 0 (1) + + β = t x ∂ ∂ ˆ Applying the following plane wave solution to Eq. (1) ' exp[ ( i kx ly t )] ψ = Ψ + − ω after substituting ' 2 2 2 2 2 and , the following ' ( / x / y ) ' v ' '/ x ς = ∇ ψ = ∂ ∂ + ∂ ∂ ψ = ∂ ψ ∂ dispersion relationship can be obtained. k β (2) u k ω = − 2 2 k l + ω β In the above, the phase speed is a function of wave number, and c u = = − k k 2 l 2 + therefore the Rossby wave is dispersive. If the basic zonal wind is zero, the phase speed of Rossby wave is always negative (westward propagating). The group velocity of the Rossby wave can be written as 2 2 ( k l ) ∂ ω β − (3) G x u = = + 2 2 2 k ( k l ) ∂ + 2 kl ∂ ω β (4) G y = = 2 2 2 l ( k l ) ∂ + β Substituting the equation of phase speed into (3), one obtains c u = − 2 2 k l + 2 2 k β . Thus, in a frame of reference which moves with the phase G x c − = 2 2 2 ( k l ) + speed c, the group velocity can be expressed as 7
� � G 2 k k l β y and (5) G = = 2 2 2 ( k l ) G k + x For stationary waves, 0 and , Eq. (2) gives c = 0 ω = β 2 2 (7) k l + = u Combination of Eqs. (3) and (4) and using Eq. (7), we obtain the equation below 2 β 2 2 2 (8) ( G u ) G u − + = = x y 2 2 2 ( k l ) + Eq. (7) shows the constraint of Rossby wave number in a steady state. The Rossby wavenumbers exist only in the circle of the k and l space shown in the figure below. From the (0, 0) source point, the particles after one second by the group velocities, Gx and Gy, should be distributed as below (RHS figure). G y l k G x u Fig. The wavenumber selection of k and l and wave front displacement (group velocity) for the stationary waves. 8
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