Why are you all so obsessed with this form-drag business? Quick-and-dirty response. I ve-been-thinking-about-this-for-a-really-long-time response

Transcription

1 Why are you all so obsessed with this form-drag business? Comments on On the Obscurantist Physics of Form Drag in Theorizing about the Circumpolar Current C. W. HUGHES Proudman Oceanographic Laboratory, Birkenhead, Merseyside, United Kingdom 29 April 1996 Quick-and-dirty response Comments on On the Obscurantist Physics of Form Drag in Theorizing about the Circumpolar Current* DIRK OLBERS Alfred-Wegener-Institute for Polar and Marine Research, Bremerhaven, Germany 16 January 1997 and 14 October 1997 I ve-been-thinking-about-this-for-a-really-long-time response

2 So what are these Obscurantist physics anyway? Time-averaged zonal momentum equation Integrate vertically over the whole water column Integrate along a streamline around the ACC ~X StressWind T z Wind stress = Topographic form stress 1 x Ap:~ fr d~ b X 0 I Ap V= ~ Oo f Ax Warren et al. treat this as a statement of mass conservation Pz oe... H... ~:- ~^ V ~t D adapted from Johnson & Bryden (1989)

3 The language obscures the more conventional physics Time-averaged zonal momentum equation Integrate vertically down to H Warren et al. treat this as a statement of mass conservation Integrate from H to D ~X StressWind T z 1 x Ap:~ fr d~ b X 0 I Ap V= ~ Oo f Ax z H Pz oe... H... ~:- ~^ V ~t D y adapted from Johnson & Bryden (1989)

4 Upwelling and buoyancy Upwelling region Johnson & Bryden claim that because there is buoyancy loss over the southern ocean, there cannot be upwelling. z H Warren et al. argue that this isn t true. This gives a surface heat loss of 1.4 x W or buoyancy loss of -0.9 x 10-7 kg m -2 s. y However, there is a large freshwater flux with buoyancy flux 2.6 x 10-7 kg m -2 s. So buoyancy is actually gained. Is this still the modern view?

5 Sverdrup dynamics set the strength of the flow Warren et al. suggest that Sverdrup dynamics dictate the strength of the ACC, with southward flow in most of the ACC and northward flow just east of Drake Passage V = r Baker (1982) found that Sverdrup balance predicts a good-ish transport of the ACC (173Sv) ACC = 1 Z C r dx Where C is a circle of constant latitude just south of Cape Horn

6 To recap Warren et al. Warren et al. have not assimilated the view that the overturning circulation is along isopycnals (a fairly recent idea at the time), so they assume diabatic flow Warren et al. assert the following The wind stress forces Ekman transport in the surface layer and the mass is returned below the topography: this process is not coupled to the transport of the ACC. The wind stress is balanced by the Coriolis force in the Ekman layer. You can predict the transport of the ACC based on Sverdrup balance alone.

7 Hughes (1996) response: Sverdrup dynamics do not provide enough constraints In a closed basin In an open channel u =0 Southward transport is returned by geostrophic flow due to slope in sea-surface height i.e. we can find u using v = fw z u x + v y + w z =0 u =0 i.e. we can add any zonally-uniform u without changing v At latitudes with no boundary, an arbitrary function of latitude can be added to the Sverdrup solution.

8 Hughes (1996) response: we can t assume that ACC transport is independent of topography Isopycnals Streamfunction (follows isopycnals) For return of the Meridional Overturning Circulation, the sloping isopycnals must reach below the topography Therefore the ACC must reach below the topography. Döös and Webb (1994)

9 Hughes (1996) response: the topography sets the strength of the eastward current Some flow appears to be returned at shallower depths than would be permitted at Drake Passage This could happen at the Kerguelen plateau But in order for this to happen you need a strong eastward current Y1 Kerguelen plateau Y2 Y4 Y3 drake passage Figure 3 Source and sink flows in a zonal channel with occluding island. Webb (1994)

10 Olbers (1998) response: here s what form drag is all about Time-averaged zonal momentum equation ]p ]t 2 f y 52 1 ]x ]z Integrate in the x and z directions E0 2 f y dz 5^t 0& 2E 2 f E2E y dz 5 0 2H 2 f E2H ]D y dz 5 pd 2 t D ]x 2D(x), Using mass conservation ^ & :5 dx, RC p D (x) 5 p(x, z 52D(x)) E0 2D(x) 7 8 y dz 5 0, ]D t 0 2 t D 1 pd 5 0, ]x 7 8

11 How does the momentum get from the surface to the bottom? Imagine two surfaces below the Ekman layer but above the level of topography b 1 b 2 Assume no diabatic transport I Z z1 I 1 Z z1 f z 2 vdzdx=0= z 2 p x dz dx

12 How does the momentum get from the surface to the bottom? ]D t 0 2 t D 1 pd 5 0, ]x 7 8 Interfacial form stress Exr 2h 1(x) ]pe ]x x 2h (x) l 2 dz dx E 2h (x) ) E [ 1 xr xr ]h 1 5 p dz 1 p(x, 2h 1(x)) ]x 2h 2(x) xl xl ]h 2 2 p(x, 2h (x)) dx. ]x 2 < 0 p < > 0 p > 0 This term is zero in a re-entrant channel

13 What is the interfacial form stress? Conservation of mass & temperature perature about a izon J 5 (J (x), J (y) ) J (z) Are heat fluxes J 5 uu 1 I J (z) 5 w u 1 I (z). Integrate over area ]w = u ]z E ]z A(z) ] ^y& 1 w da 5 0. tegral of w, which appears her Therefore [ ] ] dq 21 (y) ^y& 5 (Q ) 1 z ^yu 1 I &1 2, ]z dz ]J (z) = J 1 1 wqz 5 0. ]z zea(z) dq (y) 2Q w da 5^J &1, dz E (z) Q(z) 5 J da EA(z) (z) Is the average temperature state Is the perturbation to this state Is the diabatic heat flux Zonal momentum balance ]F 78 ]t 7 ] 8 O 2 f ^y &5 5 2 uy 1 dp, ]z ]z ]y ridges Compare with Johnson & Bryden F(f = 0) = 0 dq 21 (y) F 52f (Q ) 1 z ^yu 1 I &1 2. dz v't' P0f 0z = z~" F

14 What happens in each layer? Ekman layer 78 ]F ]t 2 f ^y &5. ]z ]z E0 2 f ^y & dz 52F (z 52E) ^t 0& 2E Intermediate layer F 2^t 0 &. Deep layer ]F O 2 f ^y &5 dp, ]z ridges E2H ]D 2 f ^y & dz 5 F (z 52H) 2 p D. 2D max 7 8 ]x

15 Why is the wind stress curl not enough? The vorticity balance y ]c 2 eπ c 1 b 5 J(p D, D) 1 curlt 0, ]x Includes the pressure at depth p D (x) 5 p(x, z 52D(x)) x = 1 = 2 The momentum equation is a boundary condition on the vorticity equation eu 1 f k 3 U 52h=p D 2 =E 1 t 0. E depends on the windstress, not the wind stress curl: RL ds =E R 5 ds [2eU2 f k 3 U 2 h=pd 1 t 0] L R 5 ds [t02 t D 1 p D=h] 5 0. L

16 Conclusions You cannot predict the transport of the ACC based on Sverdrup balance alone The momentum imparted to the ocean by the wind is transferred downwards by interfacial form stress and removed from the water column by topographic form stress. WLR never mention interfacial stress, and to some extent this s their downfall. 2F (z 52E) ^t & 0