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Fluid Dynamics of the Atmosphere and Oceans

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Fluid Dynamics of the Atmosphere and Oceans Fluid Dynamics of the Atmosphere and Oceans John Thuburn University of Exeter Introduction This module Fluid Dynamics of the Atmosphere and Oceans comprises 33 hours of lectures and examples classes. Outline of course content The equations of motion in a rotating frame; some conservation properties; circulation theorem; vorticity equation; potential vorticity equation. Hierarchies of approximate governing equations; balance and filtering. Shallow water equations: circulation and potential vorticity; energy and angular mo- mentum; gravity and Rossby waves; geostrophic balance; geostrophic adjustment; Rossby radius; quasigeostrophic shallow water equations; quasigeostrophic potential vorticity; Rossby waves; Kelvin waves. Boussinesq approximation; gravity waves in three dimensions; mountain waves; non- linear effects; eddy fluxes. Shallow atmosphere hydrostatic primitive equations, in different coordinate systems; conservation properties; Rossby and gravity waves. Quasigeostrophic theory in three dimensions: ageostrophic equations; quasigeostrophic potential vorticity equation; omega equation; free Rossby waves; Forced Rossby waves and the Charney Drazin theorem; eddy fluxes; surface waves on a potential temperature gradient; the Eady model of baroclinic instability. The planetary boundary layer; the Ekman spiral; Ekman pumping; Sverdrup balance. 1 1 Governing equations in vector form Du 1 + 2Ω u = p Φ; (1) Dt × −ρ∇ − ∇ ∂ρ + (ρu) = 0; (2) ∂t ∇ · DT p c + u = 0. (3) v Dt ρ∇ · 2 Governing equations in spherical polar coordi- nates Du uv tan φ uw 1 ∂p + 2Ωv sin φ + 2Ωw cos φ + = 0 (4) Dt − r r − ρr cos φ ∂λ Dv u2 tan φ vw 1 ∂p + + + 2Ωu sin φ + = 0 (5) Dt r r ρr ∂φ Dw u2 + v2 1 ∂p 2Ωu cos φ + g + = 0 (6) Dt − r − ρ ∂r ∂ρ + (ρu) = 0 (7) ∂t ∇ · DT p c + u = 0 (8) v Dt ρ∇ · where D ∂ u ∂ v ∂ ∂ + + + w (9) Dt ≡ ∂t r cos φ ∂λ r ∂φ ∂r 1 ∂u ∂(v cos φ) 1 ∂(r2w) u + + (10) ∇ · ≡ r cos φ ∂λ ∂φ r2 ∂r 2 3 Hierarchies of approximate equation sets (Quasi-)hydrostatic: Neglect Dw/Dt term. One of the thermodynamic • equations then effectively becomes a diagnostic equation for w, so there are really only three component prognostic equations. The resulting equations no longer support internal acoustic waves, though they do support purely horizontally propagating external acoustic waves. Scale analysis shows that the hydrostatic approximation is valid on horizontal scales large compared with a typical depth scale for the atmosphere (about 10km), or, equivalently, on timescales long compared with 1/N where N is the buoyancy frequency. Shallow atmosphere: Neglect the Coriolis terms proportional to cos φ and • certain related nonlinear terms (called the traditional approximation), and approximate r by the constant a equal to the mean radius of the Earth. These two approximations must be made together, otherwise the resulting equations fail to conserve energy and angular momentum. Hydrostatic primitive equations: Make both the hydrostatic and shallow • atmosphere approximations. Anelastic equations: Essentially, neglect ∂ρ/∂t. Then only two compo- • nents of the momentum equation can be considered as independent prognostic equations. Moreover, the momentum equations combined with the density equation imply a diagnostic equation for p and hence ρ, so there are really only three component prognostic equations. There are various versions of the anelastic equations. They do not support acoustic waves. They are valid on short horizontal length scales. Boussinesq: Assume the fluid is incompressible .u = 0, and neglect vari- • ations in density except where they appear in the∇ buoyancy term. These equations do not support acoustic waves. Quasigeostrophic: Assume hydrostatic balance, Rossby number Ro = U/(fL) • is small, temperature is not far from some reference profile, and horizontal length scale is small compared with the Earth’s radius. Usually the geometry is approximated as a β-plane. The governing equations reduce to a prognostic equation for the quasigeostrophic potential vorticity (plus another for bound- ary potential temperature), plus diagnostic equations relating the potential vorticity to winds and temperature. These equations do not support acoustic or gravity waves. Planetary geostrophic: Assume Rossby number Ro = U/(fL) is small, and • that the horizontal length scale is comparable to the Earth’s radius. Spherical geometry may be retained. The governing equations reduce to a prognostic equation for the potential vorticity, plus diagnostic equations relating the po- tential vorticity to winds and temperature. These equations do not support acoustic or gravity waves. 3 Compressible Euler equations Spherical geoid Shallow Quasi− atmosphere hydrostatic Hydrostatic shallow Anelastic atmosphere Boussinesq Shallow water Quasigeostrophic equations equations Planetary Semi− geostrophic geostrophic Quasigeostrophic shallow water equations Barotropic vorticity equation Figure 1: Part of the hierarchy of approximations to the full spherical-geometry compressible Euler equations. 4 Semi-geostrophic: Assume Rossby number Ro = U/(fL) is small, and hor- • izontal length scale is very small compared with the Earth’s radius so that the geometry can be approximated as an f-plane. These equations do not support acoustic or gravity modes. Shallow water: Assume an incompressible fluid in hydrostatic balance such • that horizontal velocity is independent of z. The resulting equations are hori- zontally two dimensional. They do not support acoustic waves but do support horizontally propagating gravity and Rossby waves. 5 4 Hydrostatic balance The following tables (based on similar tables in Holton’s book) give typical scales of terms in the governing equations for mid-latitude synoptic scale flow. Typical scales for the horizontal momentum equations u-equation Du uv tan φ uw 2Ωv sin φ 2Ωw cos φ 1 ∂p Dt r r − ρr cos φ ∂λ 2 Dv u tan φ vw 1 ∂p v-equation Dt r r 2Ωu sin φ ρr ∂φ 2 2 Scales U /L U /a W U/a f0U f0W δP/ρL Values (ms−2) 10−4 10−5 10−8 10−3 10−6 10−3 Typical scales for the vertical momentum equation 2 2 w-equation Dw u +v 2Ωu cos φ g 1 ∂p Dt − r − ρ ∂r 2 Scales UW/L U /a f0U g P0/ρH Values (ms−2) 10−7 10−5 10−3 10 10 For mid-latitude synoptic scale flow, the dominant balance in the vertical mo- mentum equation is clearly hydrostatic balance: 1 ∂p + g = 0. (11) ρ ∂z We can justify neglecting the Dw/Dt term when UW δP (12) L ≪ ρH where δP is typical horizontal variation in p. Assuming, from the horizontal mo- mentum equation, that δP δP U 2 or f UL = U 2R−1, (13) ρ ∼ ρ ∼ 0 o we require WH WH 1 or R 1. (14) UL ≪ UL o ≪ Assuming further, from the mass continuity equation, that W H W H or R , (15) U ∼ L U ∼ L o we see that hydrostatic balance will be a good approximation for H2 H2 1 or R2 1 (16) L2 ≪ o L2 ≪ where R U/(f L) is the Rossby number. o ≡ 0 6 h(x,y) u(x,y) z Figure 2: Schematic for derivation of shallow water equations. 5 The shallow water equations Although not quantitatively accurate for most atmospheric or oceanic modelling, the shallow water equations embody many of the same physical ingredients as the full governing equations, and so are valuable for developing a conceptual understanding, as well as for testing numerical integration schemes. 5.1 Derivation Consider a layer of incompressible fluid of constant density ρ in hydrostatic balance (see figure 2). Assume planar geometry and neglect 2Ω cos φ Coriolis terms. From hydrostatic balance, the pressure at any point in the fluid is given by h p(x, y, z) = ρg dz = ρg(h z), Zz − so the horizontal pressure gradient is 1 ∂p ∂h = g . ρ ∂x ∂x This is independent of z, so if u and v are initially independent of z then they will remain so. Write gh = Φ. Then the horizontal momentum equations become Du ∂Φ fv + = 0 (17) Dt − ∂x Dv ∂Φ + fu + = 0 (18) Dt ∂y For later, note the vector form: Dv + fz v + Φ = 0. (19) Dt × ∇H Take the vertical integral of the massb equation ∂u ∂v ∂w + + = 0 ∂x ∂y ∂z 7 to obtain h ∂u h ∂v dz + dz + w(h) w(0) = 0 Z0 ∂x Z0 ∂y − ∂u ∂v Dh h + h + = 0. ∂x ∂y Dt Multiplying by g gives DΦ ∂u ∂v + Φ + Φ = 0. (20) Dt ∂x ∂y or, rearranging, ∂Φ ∂ ∂ + (uΦ) + (vΦ) = 0. (21) ∂t ∂x ∂y 8 5.2 Vorticity and divergence equations Take ∂/∂x of (18) minus ∂/∂y of (17) to obtain Dζ + δζ = 0 (22) Dt or ∂ζ ∂ ∂ + (uζ) + (vζ) = 0, ∂t ∂x ∂y and take ∂/∂x of (17) plus ∂/∂y of (17) to obtain ∂δ u2 + v2 + z vζ + Φ + (23) ∂t ∇H · × ∇H 2 b where ζ = f +ξ is the absolute vorticity, ξ = ∂v/∂x ∂u/∂y is the relative vorticity, and δ = ∂u/∂x + ∂v/∂y is the divergence. − 5.3 Potential vorticity equation Combining the vorticity equation (22) with the mass equation (20) gives DΠ = 0 (24) Dt where Π = ζ/Φ. 5.4 Circulation theorem Integrating the vorticity equation within a closed material contour shows that D C = 0 (25) Dt where = vIF dl = ζ dA. C I · Z 5.5 Energy equation Take uΦ times (17) plus vΦ times (18) plus Φ+(u2 + v2)/2 times (21) to obtain ∂E ∂ uΦ2 ∂ vΦ2 + uE + + vE + = 0 (26) ∂t ∂x 2 ∂y 2 where Φ2 u2 + v2 E = + Φ . 2 2 9 5.6 Normal modes: Poincar´ewaves and Rossby waves Linearize the shallow water equations about a resting state with mean geopotential Φ0 and take f to be constant. It is easiest to work with the vorticity and divergence equations: ∂ξ + fδ = 0, ∂t ∂δ fξ + 2 Φ = 0, ∂t − ∇H ∂Φ + Φ δ = 0.
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