1 GFD – 2 Spring 2004 P.B. Rhines CHAPTER 15 [Lecture 7-03] Reading assignment: Gill Ch. 12.1 – 12.6 Rossby waves and circulation, IV: Wind-driven circulation: the forced, damped vorticity equation and Ekman layers. We now have established the basic vorticity equation of a single layer of constant density fluid, including effects bottom topography, the spherical Earth (‘β-effect’) and allowing the free-surface of the fluid to deform (Eqn. 14.5, 14.13, 14.15). To review, the original vortex stretching effect G ∂ω GGGG G abs +•∇uuω =()ωνω •∇+∇2 ∂t abs abs abs G G G ωabs =∇×u +2 Ω appears in the height variations of a fluid column, now along the local vertical (z-) axis. The β- effect relates to the change in the local vertical felt by a disk of fluid moving north or south, which has an effect similar to that of a sloping bottom: motions north or south (~ up or down the slope) cause planetary vorticity to trade with relative vorticity, and the relative vorticity creates a propagating Rossby wave. This is all summed up in the potential vorticity equation, 14.15, which is derived as follows. The vorticity and mass conservation equations for ‘shallow-water’ motions (u and v independent of z) are Df()ζ + =+()ζ f w Dt z Dh =−hu() + v = hw Dt xy z Combining to remove the horizontal divergence (or vertical stretching) term wz, Df()ζ + Dh Dh (1/) =+()ζζfh−1 =−+ () fh Dt Dt Dt (7.1) Dfζ + or ()0= Dt h which is the exact form of potential vorticity conservation neglecting forcing and viscous dissipation. The total fluid depth h includes topography and free-surface deflection, h = H + h’(x,y) + η(x,y,t); see Eqn 5.2. This potential vorticity (‘PV’) was the origin of the ‘free pathways’, the f/h contours, discussed in Chap 14. By filtering out high-frequency gravity waves we have a simpler mathematical problem for low frequencies, fT >> 1, although there is one ‘missing’ wave mode in this analysis. The Kelvin wave can have very low frequency even though it is far from being in geostrophic balance (recall that it is a long gravity wave trapped by rotation to propagate along a coast, or along the Equator). It can be generated by 2 Rossby waves in (rather rare) circumstances, for example when a Rossby wave along the Equator encounters a western boundary. Wind-stress forcing and frictional damping. We need to bring back the two terms relating to Ekman boundary layers. The complete derivation is given in small print in the succeeding section; here we summarize it. The model is one in which viscous effects, augmented by turbulent mixing, confined to thin boundary layers near the top and bottom boundaries, usually less than 25m thick. The great mass of oceanic fluid in between is very nearly immune to viscous forces. Instead of taking boundary conditions w = Dη/Dt at G top and wuh=− •∇ at the bottom, which are the inviscid (zero-viscosity) boundary conditions, we have at the top of the fluid Dη G (7.2) wzf=+ρτ−1 ˆ •∇×(/ ) (z=η-δ) Dt 0 where z=η is the position of the free surface and δ is the thickness of the Ekman layer. This rather complicated looking expression was introduced on p. 3-13. The wind-stress, G τ , is horizontal vector force acting on the sea-surface, which produces a horizontal volume flux G (7.3) −×zfτ /(ρ0 ) in the Ekman layer, and the horizontal divergence of this flux produces the vertical velocity at the base of the Ekman layer, given in Eqn. 7.2, nearly proportional to the curl of the wind-stress, yet becoming larger at low latitude where f is small (as one approaches the Equator where f=0, the theory breaks down, yet it does indicate that there will be significant upwelling due to the easterly (westward) winds there). To good approximation the boundary condition can be applied at z=0, because the thickness of the fluid layer is so much greater than δ or η. The Ekman pumping term due to the wind is fwz, with w given above. At the bottom there is a very similar Ekman layer, which acts to bring the fluid velocity smoothly to zero at the bottom. Viscous fluids require a ‘no-slip’ boundary condition (although the molecular physics of this boundary condition are a bit shaky, and in the presence of turbulence we cannot solve exactly the boundary layer flow profile). We will give a detailed derivation below, but here is an outline. The viscous stress at the G ∂u bottom is just ρν . For an Ekman layer of thickness δ, this has an approximate size ∂z ρνU/δ. The Ekman pumping velocity is the curl of this stress, which will be of approximate size G wu~/ρνδρν∇× = ζ /δ 3 and the vortex stretching term, fwz will be ~ fw/h. Thus we have a viscous damping of the vorticity which is proportional to vorticity, suggesting an exponential decay with time. The full expression is given in the derivation below. Complete derivation of the Ekman equations. We take the horizontal momentum equations for a viscous fluid, 2 −=−fvpu(1 /ρν0 ) x +∇ 2 (7.4) fupv=−(1 /ρν0 ) y + ∇ Use complex notation, so that the horizontal velocity vector is u + iv, and multiply the second equation of (7.4) by i and add to the first equation of (7.4): 2 if()(1/)() u+=− ivρν0 pxy + ip +∇+ () u iv In these thin, steadily flowing boundary layers, vertical derivatives are much greater than horizontal deriviatives, so ∇2 ≅ ∂2/∂z2. A further approximation is that the pressure gradient terms, though they vary with x and y, are vertically constant throughout the thin boundary layers. Furthermore the pressure field varies very little as one moves from the boundary layer to the geostrophic interior, again a consequence of the thinness of these layers. The pressure gradient balances the interior geostrophic flow, so it is natural to write (as in Gill, p. 320) the velocity as the sum of geostrophic flow and boundary layer flow, u + iv = (ug + ivg) + (ue + ive) where fug = -(1/ρ0)py, fvg = (1/ρ0)px. Bottom boundary layer. Consider the bottom boundary layer. For the total velocity to go to zero at the lower boundary, say at z = -H, the boundary conditions for the boundary layer part are ue + ive = 0 for z => ∞, and ue + ive = ug + ivg at z = 0. We have ∂+2 ()uiv if() u+= iv ν ee ee ∂z2 which is a simple, if unusual, o.d.e. with solutions uivuivee+=−+( gg )exp( rz ) 1+ if rifr21/2==−/;ν ( )() 2 ν (Note, z is the real vertical coordinate, not x + iy.) The horizontal velocity vector increases from zero at the boundary to match the geostrophic flow at z >> δ. Its direction wanders round, starting at 450 to the left of the geostrophic flow (northern hemisphere) at the boundary, and homing in on the geostrophic velocity at greater height. We now use this solution to give the Ekman pumping, from mass conservation: z=δ wuvdz|=− () + zH=− +δ ∫ x y zH=− The vertical integral simply gives a factor (1+i)√(f/2ν) multiplied by ∂vg/∂x -∂ug/∂y. Thus all the complicated vertical structure gives us a constant multiplier, but the interesting structure comes from the horizontal derivatives of the geostrophic velocity, in just the right combination to give the geostrophic vertical vorticity, ζ. This is a remarkably simple result: the Ekman pumping due to the boundary layer at the bottom is simply proportional to the vorticity of the flow above the boundary layer (Gill sec. 9.6): 4 w |zH=− +δ = δ eζ 1/2 δνe = (/2)f Surface Ekman layer. The same Ekman spiral solution can be turned upsidedown to give the structure of a laminar wind-driven Ekman layer, where the boundary condition is that the wind-stress match the fluid viscous stress at z approximately equal to zero. Solving the same equation with boundary conditions G G ρνuz== τ 0 GG0 z uu= g z/δ →−∞ gives uivAee+=exp( rz ) 1+ if rifr21/2==+/;ν ( )() 2 ν ρ Ai=+()ττ ν r xy The result has a velocity directed at 450 to the right of the wind-stress (northern hemisphere), with a vertically integrated volume transport lying perpendicular (right) of the stress. It is important that the real ocean and atmosphere are turbulent rather than laminar, particularly in these boundary layers. The characteristic thickness of the Ekman layer, δ = (ν/2f)1/2, is about 0.7 mm. in a laboratory setting (f = 1 sec-1, ν = 10-6 m2 sec-1), or 0.07 m for the ocean (f = 10-1 sec-1). We have no way of solving for the Ekman velocity profile in a turbulent fluid, yet the relations between external stress and Ekman volume transport (Eqn. 7.3) are valid even if the flow is turbulent; it is simply an averaged momentum balance and cannot be altered by internal stresses which redistribute the momentum. At the sea-floor it is more of a problem since we do not know the stress. Empirically, the G G stress is given by a drag coefficient multiplied by uu|| (Gill p. 328). A useful result of this kind is that the thickness of the Ekman layer in a homogeneous is approximately 0.2 u*/f, where u* is known as the ‘friction velocity’, simply the stress written in terms of a G 2 velocity, defined by ||τ = ρ0*u . 7.2 The Sverdrup interior circulation.