GFD – 2 Spring 2004 P.B. Rhines CHAPTER 15 [Lecture 7-03] Reading Assignment: Gill Ch

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 wz Dt 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 (7.1) Dt Dt Dt 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 | = δ ζ zH=− +δ e 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. The results above give us a complete vorticity equation with forcing and viscous damping, and small-height bottom topography:

22− 2 ∇η−ttxλβ η+ η −(/fHJ )(,) η h = F − R ∇η G ¤(7.5) Ffghz=•∇×(/ρτ0 ) (/) f fffυυ1 REE==δ ()2 =1/2 ( ≡ ) hhe 2 222 fH

For steady flow, with a flat bottom this is simply

2 (7.6) R∇η+β ηx =F .

This is ‘nearly’ a Helmholtz equation, and can be reduced to one by a simple change of dependent variable. Notice the similarity to the Rossby-wave equation, with ∂/∂t replaced by R. The high-order derivative in the first term suggests a boundary layer. Scale analysis of the terms in Eqn. 7.6 gives Rη/L2 βη/L F ¤The ratio of the first and second terms is R/βL. If L is the width of the ocean, R/βL will be small for realistic choices of the friction (with Ekman layer thickness small, of order 10m). In this case the first term will be negligible, leaving

βη=x F

which is known as barotropic Sverdrup balance. With ηx proportional to the meridional velocity, v, we have a remarkably simple equation relating v to the wind-forcing. The solution, (7.7) η=β −1 ∫ Fdx ¤is simple, yet we need boundary conditions in order to set the limits of integration. The v- velocity is just gF/fβ, and the u-velocity is found from mass conservation (recall that for quasi- geostrophic flow the horizontal velocity is nearly non-divergent, ux + vy = 0 + O(Ro)) , uvdx=− ∫ y which similarly needs extra arguments to settle the limits of integration. For a wind-stress curl given by F = A sin kx corresponding for example to an north-south wind that varies sinusoidally in x, the solution η = -(kβ)-1Acos kx is clear enough: an alternating pattern of north-south currents driven by north-south winds, with east-west tilt of the free surface supplying a pressure field to keep the currents in geostrophic balance. Notice that the strength of the velocity is proportional to the wind-stress, with no need for friction to limit it.

The boundary currents. Clearly there will be a difficulty if the integral of F along a latitude circle is not zero: yet, why should it be? This presented a fascinating puzzle to Henry Stommel in the late 1940s. Harald Sverdrup had articulated this interior solution (which we shall see generalizes to a stratified ocean). There are several ways to look for an answer. One is to think of the problem of a very slowly varying (in time) wind stress, and recall that low- 6 frequency, long Rossby waves radiate predominantly westward from their source. This suggests that the integral solution (7.7) at a given position should only ‘see’ forcing that lies to the east of that position. A second way to think of the problem is by solving for the Green function of Eqn. (7.6), which is the flow driven by a delta-function distribution of F. This takes a little effort but is worth it: it completely answers the question (and will be given in an optional, later section for reference). A third solution to the problem involves turning on the wind forcing from a state a rest, and watching Rossby waves set up the solution. A fourth solution to the problem is by considering an complete ocean basin with lateral boundaries and solving (7.6) with appropriate boundary conditions. In the (x,y) plane it is rather like our Ekman layer solution in the (x,z) plane, where boundary layers are invented to satisfy viscous boundary conditions.

Both of these problems are known mathematically as ‘singular perturbation problems’ in which the highest spatial derivative is multiplied by a small parameter (the viscosity). This means that an ordinary perturbation expansion in powers of ν will not work, because there are too many boundary conditions to be satisfied by the overly simple equation found for ν = 0. Texts like Bender and Orszag’s have full discussion of this interesting bit of mathematics.

Let us take Stommel’s route and solve for the complete circulation in an ocean with rectangular boundaries, at which the normal velocity vanishes. In this case we have 2 R∇η+β ηx =F The geostrophic velocity lies normal to the gradient of pressure, so our boundary condtions are ηx = 0 on y = 0 and y=Ly, and ηy = 0 on x = 0, Lx (more generally, ∇η×n = 0 on the boundary, n being a unit vector normal to the boundary). This just says that η is constant (or at most a function of time) around the boundary; the boundary is a streamline.

For small values of R (better described with a non-dimensional number, R/βL << 1), the Sverdrup balance holds in most of the fluid yet we will need boundary layers. Near the 2 boundary, we know that R∇ η will be large, and it will then tend to be balanced by β ηx rather than F. This happens because the x-derivative will make βηx bigger than typical values of F, which are in balance with the much smaller βηx values found in the interior far from boundaries. Thus we have 2 R∇η+β ηx =0 in the vicinity of the boundaries. Consider a north-south boundary at which η takes a constant value. The east-west variation of η far exceeds the north-south variation in a thin region near the boundary, hence the equation becomes

Rη+η=xx β x 0 which integrates once with respect to x, and then yields exponential solutions, η=NxRexp( −β / ) 7 for N a constant. Notice the sign; it is a first-order equation and hence there is just one exponential solution, not two. The solution decays eastward and hence this can only be a western boundary current. The full solution for this problem (valid for small R/βLx ) is Lx η=(1 − exp( −x /δβ )) −1 Fdx S ∫ (7.8) x

δβS = R / The choice of limits of integration allow η to vanish on the eastern boundary, while the boundary layer allows η to vanish on the western boundary. The solution at a given longitude, x, responds to wind-stress curl forcing only to the east of it. This is what we expect from the predominantly westward propagation of Rossby waves, which ‘carry the information’ about the forcing. If we are lucky, and F vanishes at the northern and southern boundaries, then this is

¤the complete solution. If not we have to add boundary layers at those walls too. δS is known as the Stommel boundary-layer width, and it is a model of the Gulf Stream, the Kuroshio, and the other western boundary currents found in the major ocean basins.

Next we will discuss the development of the circulation from a state of no motion, when the wind begins to blow at time t=0 (Gill 12.4). 8

Figure 7.1 From Harrison, J. Climate 1989, giving an estimate of the annual-average wind- stress acting on the ocean. 9

Fig. 7.2 From Harrison, J Climate 1989, Giving two estimated maps of the Sverdrup transport (the x-integral of the curl of the wind-stress, starting at the eastern boundary and 10 integrating west). The lower is from Hellerman and Rosenstein’s classic calculation, while the upper is newly estimated (as of 1989) from weather-center data.

Fig. 7.3. Winds on 28-29 Jan 2003 from SeaWinds, the latest in the family of scatterometer satellites using the roughening of the sea-surface by winds to give quantitative wind fields. The satellite covers 90% of the Earth’s surface in one day (note data gaps between swaths), so that while some synoptic information is missing from the quickly moving wind patterns, there is much fine detail in the maps even before time- averaging.

Fig. 7.4a From Chelton, 2003. Map of 3-year averaged windstress curl from scatterometer satellite observations (1999-2002). There are many fine-scale features associated with ocean fronts and bottom topography (Southern Ocean) and wind shadows, equatorial winds, the wakes of topography like Greenland.

Fig. 7.4b Wind-stress from QuikSCAT (left) and from NCEP weather grid (based on radiosonde pressure with some assimilated extra information like cloud-movement velocities and to some extent QuikSCAT itself. An intense cyclonic storm draws air along the Labrador Sea from the northwest. Color shows intensity of wind-stress. The satellite has enough passes to make near-synoptic maps like this. 13

Fig. 7.5 Depth integrated transport (in Sverdrups (106 m3 sec-1)). Integration is from zero at the coast of Antarctica reaching 130 Sverdrups along the American coast and Greenland. Along the coasts of Europe and Africa it reaches 132 Sverdrups as a net of 2 Sverdrups are assumed entering the Atlantic from the Arctic. Shaded area represents depths less than 3500m. Notice the western boundary currents, the northern boundary current near Greenaland, and the Antarctic Circumpolar Current which veers northward after passing round the tip of South America.

from Reid, J.R. On the total geostrophic circulation of the North Atlantic Ocean: flow patterns, tracers and transports. Prog. Oceanography 33, 1994

Fig. 7.6 Estimated flow at 1000 db and 1500 db (Reid, 1994), expressed as a steric height anomaly D (in units of 102 m2 sec-2). This quantity is based on the thermal wind equation, whereby the horizontal velocity can be expressed by the geostrophic pressure field, which acts as a streamfunction. The pressure is the vertical integral of the hydrostatic equation from some reference level. This integral of the density field is related to the height of a -1 column of fluid compared with a reference column. The result is that f (Dy, -Dx ) = (u,v), so that the contours plotted can be turned into velocities.