Fundamentals Of Geophysical Fluid Dynamics Part 2 ppt

58 Fundamental Dynamics x, u X, U y, v Y, V z, w Z, W Ω t & T & rotating coordinatesstationary coordinates Fig. 2.9. A rotating coordinate frame with coordinates, (X, Y, Z, T ), and a non-rotating frame with coordinates, (x, y, z, t). The rotation vector is parallel to the vertical axis, ΩΩΩ = Ω ˆ z. = D Dt r = ∂ T + U∂ X + V ∂ Y + W ∂ Z . (2.94) ∇∇∇ s = ˆ x∂ x + ˆ y∂ y + ˆ z∂ z = ∇∇∇ r = ˆ X∂ X + ˆ Y∂ Y + ˆ Z∂ Z . (2.95) Similarly, the incompressible continuity equation in (2.38) preserves its form, ∇∇∇ s · u = ∇∇∇ r · U = 0 , (2.96) implying that material parcel volume elements are the same in each frame, with dx = dX. The tracer equations in (2.38) also preserve their form because of (2.94). The material acceleration transforms as Du Dt s = D Dt [ ˆ xu + ˆ yv + ˆ zw] = D Dt [ ˆ X(U −ΩY ) + ˆ Y(V + ΩX) + ˆ ZW ] = DU Dt r + 2Ω ˆ Z ×U + 1 ρ 0 ∇∇∇ r P , (2.97) with P = − ρ 0 Ω 2 2 (X 2 + Y 2 ) . (2.98) 2.4 Earth’s Rotation 59 The step from the first and second lines in (2.97) is an application of (2.92). In the step to the third line, use is made of (2.94) and the relations, D ˆ X Dt = Ω ˆ Y, D ˆ Y Dt = −Ω ˆ X, D ˆ Z Dt = 0 , (2.99) that describe how the orientation of the transformed coordinates ro- tates. Since ∇∇∇ s φ = ∇∇∇ r φ by (2.95), the momentum equation in (2.38) transforms into DU Dt r + 2Ω ˆ Z ×U = −∇∇∇ r  φ + P ρ 0  − ˆ Z gρ ρ 0 + F . (2.100) After absorbing the incremental centrifugal force potential, P/ρ 0 , into a redefined geopotential function, φ, then (2.100) has almost the same mathematical form as the original non-rotating momentum equation, al- beit in terms of its transformed variables, except for the addition of the Coriolis force, −2ΩΩΩ × U. The Coriolis force has the effect of accelerat- ing a rotating-frame horizontal parcel displacement in the horizontally perpendicular direction (i.e., to the right when Ω > 0). This acceler- ation is only an apparent force from the perspective of an observer in the rotating frame, since it is absent in the inertial-frame momentum balance. Hereafter, the original notation (e.g., x) will also be used for rotating coordinates, and the context will make it clear which reference frame is being used. Alternative geometrical and heuristic discussions of this transformation are in Pedlosky (Chap. 1.6, 1987), Gill (Chap. 4.5, 1982), and Cushman-Roisin (Chap. 2, 1994). 2.4.2 Geostrophic Balance The Rossby number, Ro, is a non-dimensional scaling estimate for the relative strengths of the advective and Coriolis forces: u · ∇∇∇u 2ΩΩΩ × u ∼ V V /L 2ΩV = V 2ΩL , (2.101) or Ro = V fL , (2.102) where f = 2Ω is the Coriolis frequency. In the ocean mesoscale eddies and strong currents (e.g., the Gulf Stream) typically have V ≤ 0.5 m s −1 , L ≈ 50 km, and f ≈ 10 −4 s −1 (∼ 2π day −1 ); thus, Ro ≤ 0.1. 60 Fundamental Dynamics In the atmosphere the Jet Stream and synoptic storms typically have V ≤ 50 m s −1 , L ≈ 10 3 km, and f ≈ 10 −4 s −1 ; thus, Ro ≤ 0.5. Therefore, large-scale motions have moderate or small Ro, hence strong rotational influences on their dynamics. Motions on the planetary scale have a larger L ∼ a and usually a smaller V , so their Ro values are even smaller. Assume as a starting model the rotating Primitive Equations with the hydrostatic approximation (2.58). If t ∼ L/V ∼ 1/fRo, F ∼ RofV (or smaller), and Ro  1, then the horizontal velocity is approximately equivalent to the geostrophic velocity, u g = (u g , v g , 0), viz., u h ≈ u g , and the horizontal component of (2.100) becomes fv g = ∂φ ∂x , fu g = − ∂φ ∂y , (2.103) with errors O(Ro). This is called geostrophic balance, and it defines the geostrophic velocity in terms of the pressure gradient and Coriolis frequency. The accompanying vertical force balance is hydrostatic, ∂φ ∂z = −g ρ ρ o = −g(1 −αθ) , (2.104) expressed here as a notational hybrid of (2.33), (2.58) and (2.80) with the simple equation of state, ρ/ρ o = 1 −αθ . Combining (2.103)-(2.104) yields f ∂v g ∂z = gα ∂θ ∂x , f ∂u g ∂z = −gα ∂θ ∂y , (2.105) called thermal-wind balance. Thermal-wind balance implies that the ver- tical gradient of horizontal velocity (or vertical shear) is directed along isotherms in a horizontal plane with a magnitude proportional to the horizontal thermal gradient. Geostrophic balance implies that the horizontal velocity, u g , is ap- proximately along isolines of the geopotential function (i.e., isobars) in horizontal planes. Comparing this with the incompressible velocity po- tential representation (Sec. 2.2.1) shows that ψ = 1 f φ + O(Ro) ; (2.106) 2.4 Earth’s Rotation 61 i.e., the geopotential is a horizontal streamfunction whose isolines are streamlines (Sec. 2.1.1). For constant f (i.e., the f-plane approxi- mation), ∂ x u g + ∂ y v g = 0 for a geostrophic flow; hence, δ h = 0 and w = 0 at this order of approximation for an incompressible flow. So there is no divergent potential as part of the geostrophic velocity, i.e., X = χ = 0 (Sec. 2.2.1). However, the dynamically consistent evo- lution of a geostrophic flow does induce small but nonzero X, χ, and w fields associated with an ageostrophic velocity component that is an O(Ro) correction to the geostrophic flow, but the explanation for this is deferred to the topic of quasigeostrophy in Chap. 4. Now make a scaling analysis in which the magnitudes of various fields are estimated in terms of the typical magnitudes of a few primary quan- tities plus assumptions about what the dynamical balances are. The way that it is done here is called geostrophic scaling. The primary scales are assumed to be u, v ∼ V , x, y ∼ L , z ∼ H , f ∼ f 0 . (2.107) From these additional scaling estimates are derived, T ∼ L V , p ∼ ρ 0 f 0 V L , ρ ∼ ρ 0 f 0 V L gH , (2.108) by advection as the dominant rate for the time evolution, geostrophic balance, and hydrostatic balance, respectively. For the vertical velocity, the scaling estimate from 3D continuity is W ∼ V H/L. However, since geostrophic balance has horizontal velocities that are approximately hor- izontally non-divergent (i.e., ∇∇∇ h ·u h = 0), they cannot provide a balance in continuity to a w with this magnitude. Therefore, the consistent w scaling must be an order smaller in the expansion parameter, Ro, viz., W ∼ Ro V H L = V 2 H f 0 L 2 . (2.109) Similarly, by assuming that changes in f(y) are small on the horizontal scale of interest (cf., (2.88)) so that they do not contribute to the leading- order momentum balance (2.103), then β = df dy ∼ Ro f 0 L = V L 2 . (2.110) This condition for neglecting β can be recast, using β ∼ f 0 /a (with a ≈ 6.4×10 6 m, Earth’s radius), as a statement that L/a = Ro  1, i.e., L is a sub-global scale. Finally, with geostrophic scaling the condition 62 Fundamental Dynamics for validity of the hydrostatic approximation in the vertical momentum equation can be shown to be Ro 2  H L  2  1 (2.111) by an argument analogous to the non-rotating one in Sec. 2.3.4 (cf., (2.72)). Equipped with these geostrophic scaling estimates, now reconsider the basis for the oceanic rigid-lid approximation (Sec. 2.3.3). The approxi- mation is based on the smallness of D t h compared to interior values of w. The scalings are based on horizontal velocity, V , horizontal length, L, vertical length, H, and Coriolis frequency, f, a geostrophic estimate for the sea level fluctuation, h ∼ fV L/g, and the advective estimate, D t ∼ V/L. These combine to give D t h ∼ fV 2 /g. The geostrophic estimate for w is (2.109). So the rigid lid approximation is accurate if w  Dh Dt V 2 H f 0 L 2  fV 2 g R 2 e  L 2 , (2.112) with R e = √ gH f . (2.113) R e is called the external or barotropic deformation radius (cf., Chap. 4), and it is associated with the density jump across the oceanic free surface (as opposed to the baroclinic deformation radii associated with the interior stratification; cf., Chap. 5). For mid-ocean regions with H ≈ 5000 m, R e has a magnitude of several 1000s km. This is much larger than the characteristic horizontal scale, L, for most oceanic currents. Geostrophic scaling analysis can also be used to determine the con- ditions for consistently neglecting the horizontal component of the local rotation vector, f h = 2Ω e cos[θ], compared to the local vertical compo- nent, f = 2Ω e sin[θ] (Fig. 2.8). The Coriolis force in local Cartesian coordinates on a rotating sphere is 2ΩΩΩ e × u = ˆx (f h w − fv) + ˆy f u − ˆz f h u . (2.114) In the ˆx momentum equation, f h w is negligible compared to fv if Ro H L f h f  1 , (2.115) 2.4 Earth’s Rotation 63 based on the geostrophic scale estimates for v and w. In the ˆz momentum equation, f h u is negligible compared to ∂ z p/ρ 0 if H L f h f  1 , (2.116) based upon the geostrophic pressure scale, p ∼ ρ 0 fLV . In middle and high latitudes, f h /f ≤ 1, but it becomes large near the Equator. So, for a geostrophic flow with Ro ≤ O(1), with small aspect ratio, and away from the Equator, the dynamical effect of the horizontal component of the Coriolis frequency, f h , is negligible. Recall that thinness is also the basis for consistent hydrostatic balance. For more isotropic motions (e.g., in a turbulent Ekman boundary layer; Sec. 6.1) or flows very near the Equator, where f  f h since θ  1, the neglect of f h is not always valid. 2.4.3 Inertial Oscillations There is a special type of horizontally uniform solution of the rotating Primitive Equations (either stably stratified or with uniform density). It has no pressure or density variations around the resting state, no vertical velocity, and no non-conservative effects: δφ = δθ = w = F = Q = ∇∇∇ h = 0 . (2.117) The horizontal component of (2.100) implies ∂u ∂t − fv = 0, ∂v ∂t + fu = 0 , (2.118) and the other dynamical equations are satisfied trivially by (2.117). A linear combination of the separate equations in (2.118) as ∂ t (1st) + f × (2nd) yields the composite equation, ∂ 2 u ∂t + f 2 u = 0 . (2.119) This has a general solution, u = u 0 cos[ft + λ 0 ] . (2.120) Here u 0 and λ 0 are amplitude and phase constants. From the first equation in (2.118), the associated northward velocity is v = −u 0 sin[ft + λ 0 ] . (2.121) 64 Fundamental Dynamics The solution (2.120)-(2.121) is called an inertial oscillation, with a pe- riod P = 2π/f ≈ 1 day, varying from half a day at the poles to infinity at the Equator. Its dynamics is somewhat similar to Foucault’s pendu- lum that appears to a ground-based observer to precess with frequency f as Earth rotates underneath it; but the analogy is not exact (Cushman- Roisin, Sec. 2.5, 1994). Durran (1993) interprets an inertial oscillation as a trajectory with constant absolute angular momentum about the axis of rotation (cf., Sec. 3.3.2). For such a solution, the streamlines are parallel, and they rotate clockwise/counterclockwise with frequency |f | for f > 0/< 0 in the northern/southern hemisphere when viewed from above. The associ- ated streamfunction (Sec. 2.2.1) is ψ(x, y, t) = −u 0 (x sin[ft + λ 0 ] + y cos[ft + λ 0 ]) . (2.122) The trajectories are circles (going clockwise for f > 0) with a radius of u 0 /f, often called inertial circles. This direction of rotary motion is also called anticyclonic motion, meaning rotation in the opposite direction from Earth’s rotation (i.e., with an angular frequency about ˆ z with the opposite sign of f). Cyclonic motion is rotation with the same sign as f. The same terminology is applied to flows with the opposite or same sign, respectively, of the vertical vorticity, ζ z , relative to f (Chap. 3). Since f ∼ Ω ≈ 10 −4 s −1 , it is commonly true that f  N in the atmo- spheric troposphere and stratosphere and oceanic pycnocline. Inertial oscillations are typically slower than buoyancy oscillations (Sec. 2.2.3), but both are typically faster than the advective evolutionary rate, V/L, for geostrophic winds and currents. 3 Barotropic and Vortex Dynamics The ocean and atmosphere are full of vortices, i.e., locally recirculating flows with approximately circular streamlines and trajectories. Most of- ten the recirculation is in horizontal planes, perpendicular to the gravita- tional acceleration and rotation vectors in the vertical direction. Vortices are often referred to as coherent structures, connoting their nearly uni- versal circular flow pattern, no matter what their size or intensity, and their longevity in a Lagrangian coordinate frame that moves with the larger-scale, ambient flow. Examples include winter cyclones, hurricanes, tornadoes, dust devils, Gulf-Stream Rings, Meddies (a sub-mesoscale, subsurface vortex, with L ∼ 10s km, in the North Atlantic whose core water has chemical properties characteristic of the Mediterranean out- flow into the Atlantic), plus many others without familiar names. A coincidental simultaneous occurrence of well-formed vortices in Davis Strait is shown in Fig. 3.1. The three oceanic anticyclonic vortices on the southwestern side are made visible by the pattern of their advec- tion of fragmentary sea ice, and the cyclonic atmospheric vortex to the northeast is exposed by its pattern in a stratus cloud deck. Each vortex type probably developed from an antecedent horizontal shear flow in its respective medium. Vortices are created by a nonlinear advective process of self-organization, from an incoherent flow pattern into a coherent one, more local than global. The antecedent conditions for vortex emergence, when it occurs, can either be incoherent forcing and initial conditions or be a late-stage outcome of the instability of a prevailing shear flow, from which fluctua- tions extract energy and thereby amplify. This self-organizing behavior conspicuously contrasts with the nonlinear advective dynamics of tur- bulence. On average turbulence acts to change the flow patterns, to increase their complexity (i.e., their incoherence), and to limit the time 65 66 Barotropic and Vortex Dynamics Fig. 3.1. Oceanic and atmospheric vortices in Davis Strait (north of the Labrador Sea, west of Greenland) during June 2002. Both vortex types are mesoscale vortices with horizontal diameters of 10s-100s km. (Courtesy of Jacques Descloirest, NASA Goddard Space Flight Center.) over which the evolution is predictable. A central problem in GFD is how these contrasting paradigms — coherent structures and turbulence — can each have validity in nature. This chapter is an introduction to these phenomena in the special situation of two-dimensional, or barotropic, fluid dynamics. 3.1 Barotropic Equations Consider two-dimensional (2D) dynamics, with ∂ z = w = δρ = δθ = 0, and purely vertical rotation with ΩΩΩ = ˆ z f/2. The governing momentum and continuity equations under these conditions are Du Dt − fv = − ∂φ ∂x + F (x) Dv Dt + fu = − ∂φ ∂y + F (y) 3.1 Barotropic Equations 67 ∂u ∂x + ∂v ∂y = 0 , (3.1) with D Dt = ∂ ∂t + u ∂ ∂x + v ∂ ∂y . These equations conserve the total kinetic energy, KE = 1 2   dx dy u 2 , (3.2) when F is zero and no energy flux occurs through the boundary: d dt KE = 0 (3.3) (cf., Sec. 2.1.4 for constant ρ and e in 2D). Equation (3.3) can be derived by multiplying the momentum equation in (3.1) by u· , integrating over the domain, and using continuity to show that there is no net energy source or sink from advection and pressure force. The 2D incompress- ibility relation implies that the velocity can be represented entirely in terms of a streamfunction, ψ(x, y, t), u = − ∂ψ ∂y , v = ∂ψ ∂x , (3.4) since there is no divergence (cf., (2.24)). The vorticity (2.26) in this case only has a vertical component, ζ = ζ z : ζ = ∂v ∂x − ∂u ∂y = ∇ 2 ψ . (3.5) (In the present context, it is implicit that ∇∇∇ = ∇∇∇ h .) There is no buoy- ancy influence on the dynamics. This is an example of barotropic flow using either of its common definitions, ∂ z = 0 (sometimes enforced by taking a depth average of a 3D flow) or ∇∇∇φ × ∇∇∇ρ = 0. (The opposite of barotropic is baroclinic; Chap. 5). The consequence of these simplifying assumptions is that the gravitational force plays no overt role in 2D fluid dynamics, however much its influence may be implicit in the rationale for why 2D flows are geophysically relevant (McWilliams, 1983). 3.1.1 Circulation The circulation (defined in Sec. 2.1) has a strongly constrained time evolution. This will be shown using an infinitesimal calculus. Consider the time evolution of a line integral  C A · dr, where A is an arbitrary [...]... ∂u ∂2u ∂2v ∂ζ −→ − + = ; ∂t ∂y∂t ∂x∂t ∂t (u · )u −→ = = ∂u ∂ ∂v ∂u ∂v ∂ u + u +v +v ∂y ∂x ∂y ∂x ∂x ∂y ∂2v ∂2u ∂2u ∂2v +v − − 2 u 2 ∂x ∂y∂x ∂y∂x ∂y ∂u ∂u ∂v ∂v ∂u ∂v − + + + ∂y ∂x ∂y ∂x ∂x ∂y u· ζ , (3.19) − (3 .20 ) using the 2D continuity relation in (3.1); φ −→ − ˆ fz × u ∂ ∂y ∂φ ∂x ∂ ∂x ∂φ ∂y = 0; ∂ ∂ (−f v) + (f u) ∂y ∂x ∂u ∂v ∂f ∂f f +u + +v ∂x ∂y ∂x ∂y −→ − = + (3 .21 ) 72 Barotropic and Vortex Dynamics. .. the existence of another solution with the opposite parity with the direction of motion in y reversed In general 2D dynamics is parity invariant, even though the divergence relation and its associated pressure field are not parity invariant Specifically, the 2D dynamics of cyclones and an- 3.1 Barotropic Equations 79 V fr /2 C A − fr /2 1 ∂φ  f ∂r fr /2 − fr /2 B C Fig 3.4 Graphical solution of axisymmetric... the form of H(x ) in (3.69) is unchanged from H(x) since |xα − xβ | = |xα − xβ | Thus, there are 4 integrals of the motion and 2N degrees of freedom (i.e., the (x α , yα )) in the dynamical system, hence 2N − 4 independent degrees of freedom in a point-vortex system There are infinitely more integral invariants of conservative, 2D fluid 3 .2 Vortex Movement 85 dynamics that are the counterparts of G[q]...68 Barotropic and Vortex Dynamics v2 ∆ t r2 ∆r r1 ∆r r2 r 1 v1∆ t C at t = t + ∆ t C at t Fig 3 .2 Schematic of circulation evolution for a material line that follows the flow C is the closed line at times, t and t = t + ∆t The location of two neighboring points are r1 and r2 at time t, and they move with velocity v1 and v2 to r1 and r2 at time t vector and C is a closed material... invariance of H is an expression of energy conservation in the point-vortex approximation to the general expression for 2D fluid dynamics (3.3) Other integral invariants of (3.63)-(3.69) are X ≡ Y ≡ I ≡ Cα Cα x α / α α Cα Cα y α / α α 2 Cα (x2 + yα ) α (3.71) α This can be verified by taking the time derivative and substituting the equations of motion These quantities are point-vortex counterparts of the... φ ∼ f0 V L, β = (3.41) 1, (3.39) becomes 2 2 φ = f0 ψ[ 1 + O(Ro) ] =⇒ This is the geostrophic balance relation (2. 106) For general f and Ro, the 2D divergence equation is 2 φ = · (f ψ) + 2J ∂ψ ∂ψ , ∂x ∂y , (3. 42) again neglecting · F This is called the gradient-wind balance relation Equation (3. 42) is an exact relation for conservative 2D motions, but also often is a highly accurate approximation for... operator vanishes if each of its arguments is a function of a single variable, here y In a zonal flow, all other flow quantities (e.g., ψ, φ, ζ, q) are functions only of the coordinate y Vortex Flow: A simple example of a vortex solution for 2D, conservative, uniformly rotating (i.e., f = f0 ) dynamics is an axisymmetric flow where ψ(x, y, t) = ψ(r), and r = [(x − x0 )2 + (y − y0 )2 ]1 /2 is the radial distance... parcels along it) A small increment along the curve between two points marked 1 and 2, ∆r = r2 − r1 , becomes ∆r after a small interval, ∆t (Fig 3 .2) : ∆r ≡ ≈ = r2 − r1 (r2 + u2 ∆t) − (r1 + u1 ∆t) ∆r + (u2 − u1 )∆t , (3.6) using a Taylor series expansion in time for the Lagrangian coordinate, r(t) Thus, ∂u ∆r − ∆r ≈ u2 − u1 ≈ ∆s = (∆r · )u ∆t ∂s (3.7) 3.1 Barotropic Equations 69 for small ∆s = |∆r|,... Vortex Dynamics u· f ; = (3 .22 ) and F −→ − ∂F x ∂F y + = F ∂y ∂x (3 .23 ) The result is Dζ = −u · f + F Dt (3 .24 ) The vorticity only changes following a parcel because of a viscous or external force curl, F, or spatial variation in f Notice the similarity with Kelvin’s theorem (3.16) This is to be expected because, as derived in Sec 2. 1, C u · dr = ζ dx dy (3 .25 ) A Equation (3 .24 ) is a local differential... Hamiltonian functional for point vortices defined by H(pα , qα ) = − 1 2 α,β Cα Cβ ln |xα − xβ | (3.69) 84 Barotropic and Vortex Dynamics The primed summation again excludes all terms with α = β As is often true in Hamiltonian mechanics, H is interpreted as the energy of the system In 2D dynamics the only type of energy is the kinetic energy, KE (3 .2) In this context H is called the interaction kinetic energy . ] = DU Dt r + 2 ˆ Z ×U + 1 ρ 0 ∇∇∇ r P , (2. 97) with P = − ρ 0 Ω 2 2 (X 2 + Y 2 ) . (2. 98) 2. 4 Earth’s Rotation 59 The step from the first and second lines in (2. 97) is an application of (2. 92) . In. D t h ∼ fV 2 /g. The geostrophic estimate for w is (2. 109). So the rigid lid approximation is accurate if w  Dh Dt V 2 H f 0 L 2  fV 2 g R 2 e  L 2 , (2. 1 12) with R e = √ gH f . (2. 113) R e is. in (2. 118), the associated northward velocity is v = −u 0 sin[ft + λ 0 ] . (2. 121 ) 64 Fundamental Dynamics The solution (2. 120 )- (2. 121 ) is called an inertial oscillation, with a pe- riod P = 2 /f