QUASI-GEOSTROPHIC DYNAMICS OF DENSE WATER ON SLOPING TOPOGRAPHY How does topography impact deep ocean circulation?

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1 QUASI-GEOSTROPHIC DYNAMICS OF DENSE WATER ON SLOPING TOPOGRAPHY How does topography impact deep ocean circulation? Anna Wåhlin Department of Earth Sciences University of Gothenburg (Sweden)

2 Density Currents Formation

3 1.5-layer system: Well-mixed in all directions Dense layer active Upper layer at rest Upper layer infinitely deep (hence, no velocities even though pushed by lower layer

4 Outline: u u u + u + v fv = g' ( h+ D) F t x y x h X 4 Today: h uh vh + + = t x y 0 1. Geostrophic adjustment on sloping topography (2 & 3; constant slope) 2. Ekman dynamics (2 & 3 & 4; constant slope) 3. Tuesday: Topographic channeling (2 & 3 and 4; steady state; small-scale topography)

5 Geostrophic adjustment: Bottom currents tend to flow along steep bathymetry. (Dense and intermediate water masses in the Arctic ocean) Canadian Basin Deep Water Intermediate water masses From Björk et al, DSR 2010

6 Clarification (analogy topographic and planetary β): Planetary β: Sloping topography: f f y = 0 + β 0 D= D +α x fu = g ' ( h + D) y fv = g ' ( h + D) x g' g' g' h+ hh + h h hh + t y x xy y x f0 f0 f0 g' g' + h( βh hxy ) = 0 f 2 x 0 f0 h g' h βh = 0 t f x 2 0 h + ( uh) + ( vh) = 0 t x y ' ' ' ' ht + g hyhx + h g h g xy hy ( hx + α) h g hxy = 0 f f f f h g' α h = 0 t f y

7 Ekman dynamics: Frictional smoothing in interior Lower boundary creeps downward creating a thin layer Upper boundary becomes horizontal

8 Laminar regime (deep ambient water, small source salinity Shallow level part Dense source Straight slope Exchangeable section Deep level part

9 Wave regime (Cenedese et al) (shallow ambient water, large source salinity

10 Eddy regime (shallow ambient water, small source salinity

11 Constant slope: Only small downward transport Shallow Main flow Frictional transport Eddy-induced transport Deep From Cenedese et al, 2004

12 Field of small-scale corrugations on the Ross Sea slope From Muench et al, to be submitted

13 Amundsen Sea

14 Large part of the World s continental slopes are corrugated: # canyons per 100 km From Allen and de Madron, 2009

15 Downslope flow in numerical simulation [nek5000 model: Özgökmen et al., 2004; Özgökmen & Fischer, 2008] Downslope flow is enhanced by the corrugations, and increasingly so by the shorter wavelengths. Smooth bottom (no corrugations) H C =200 m λ C =2 km H C =200 m λ C =0.7 km From Muench et al, to be submitted Normalized density

16 Downslope flow as a function of height H C at a fixed (2 km) wavelength λ C (GOLD model) Smooth bottom (no corrugations) H C =20 m λ C =2 km H C =100 m λ C =2 km From Muench et al, to be submitted

17 Submarine ridge intersects the slope: Ridge Shallow Deep From Darelius (DSR 2008)

18 Submarine canyon intersects the slope Deep Shallow Canyon

19 Geostrophic flow along the depth contours, out of the canyon area:

20 Filchner Overflow (Weddel Sea, Antarctica): Darelius et al, Tellus, 2008

21 Instead of flowing out of the canyon area, dense water channeled downhill, trapped in corrugations: Something counteracts the geostrophic tendency

22 Inside the canyon/corrugation: Flow downhill inside the canyon induces Ekman transport to the left, i.e. opposing the geostrophic flow out of the canyon From Davies et al, 2006

23 Inside the canyon/corrugation: When downhill flow is sufficiently fast, the two crosscanyon flows (geostrophic and Ekman) cancel. h V + δ V =0 g Can be solved. V g & V E found assuming balance between pressure gradient, Coriolis force & friction in both directions E

24 View of the circulation: Up canyon From above From Darelius (DSR 2008)

25 Mathematical description 1.5-layer system: Well-mixed in all directions Dense layer active Upper layer at rest Upper layer infinitely deep Steady state Topography: D( x, y) = D + α x + d( y) 0 Constant slope in x- direction topographic feature in y- direction (canyon, ridge, corrugation)

26 Mathematical description K fv = g ' α u fu g'( hy Dy) h = + K v h hv g ' α K '( = h + y + y) u u f f f g h D K v = f h δ L vh K = f δ = = + h 2 1 (1 + ) δ ( ) 2 LuG + vh G ql h δ LuG vg ( ) ( ) h

27 Mathematical description hv hv = q ( h δ u + vh L ) L G G ( ) = 0 h= 0 or hv + δu = 0 G G h g ' α g' ( h D) ( h D) δ + = 0 hα = δ + f f y y α = slope of canyon axis δ = thickness of frictional boundary layer

28 Mathematical description ( h D) hα = δ + y ( hd, ) = D( hd ˆ, ˆ ) y = Wyˆ C h hs D hˆ ˆ ˆ D = = βh y δ y yˆ yˆ β = αw δ α = slope of canyon/channel axis W = canyon width D C = canyon/channel depth δ = thickness of frictional boundary layer

29 Cosine-shaped channel: h = h + h h h P H P H DC y y = cos( ) + β sin( ) 2 2(1 + β ) W W = Ce β y W Dependence on β β = 0.1 β = αw δ DC y Dy ( ) = (1 cos( )) 2 W β =1 β =10

30 Darelius&W, 2007 Different shapes ridges, 3 different β:

31 β = αw δ α = slope of canyon/channel axis δ= frictional boundary layer thickness W = canyon/channel width Muench et al, JGR 2009

32 Dimensional solution (change β by changing width) Small transport β = αw δ u = 0 at edges (slippery boundary layers) Large transport From Davies et al, 2006

33 Cross-section looking uphill inside the canyon: Generally good agreement in laboratory experiments From Darelius (DSR 2008) From Davies et al, JPO 2006

34 Cosine-shaped channel: Dependence on BC h= h + h h h P H P H DC y y = cos( ) + β sin( ) 2 2(1 + β ) W W = C y W e β C = 0 C > 0 β =1 C < 0

35 Many different solutions that have zero cross-canyon transports. All with different along-canyon transport. Can be used as boundary condition: More than Qmax => spills over edge, continues along the slope (or into next canyon) Q MAX Transport capacity Q MAX : the channel is completely filled

36 Transport capacity: ˆ Q ˆ ˆ MAX = hu dy = dy = h dy = h D dy yˆ R R R 2 Rˆ ' ' ' C G g D g D gd h h + f y f y f L L L Lˆ QMAX 2 gd ' C = r( β) = QBCr( β) f β = αw δ Only a function of β

37 QMAX Largest Qmax for β~1 (depending on friction parameterization) 2 gd ' C = r( β) = QBCr( β) f r( β ) β = αw δ 2Q Q BC β

38 r(β) for ridges and canyons Q Q BC Darelius & W, 2007 β

39 Deep canyon Shallow canyon

40 Q > Qmax => spills over edge, continues along the slope (or into next canyon) From Darelius, DSR 2008

41 Experiments with sloping channels: Generally good agreement in laboratory experiments From Darelius (DSR 2008) From Davies et al, JPO 2006

42 Experiments with sloping channels: U H α

43 Non-rotating channel flow (Linden, others ): Fr gh ' = = 2 U Fr C Adapt non-rotating to rotating (Koman, 1969): H H ' FrC = U = g H 4 3 Q = UHW g ' dh fu dy = W => U and H are constants and equal to U gh ' WFr = 3 = C H Q WU

44 Compare to Ekman dynamics model: H h g ' α g ' ( h + D) δ = f f y 0 H h hs D h hα = = y δ y y δ W h( y) = Ce α y δ W W g' h g' ( 2 2 G R L ) f y f 0 0 Q = hu dy = h dy = h h H = h h R L

45 From Cossu et al, submitted

46 Comparing velocities B gh ' = 1 f W Koman 2 B >1 2 B <1 From Cossu et al, submitted

47 H B gh ' = 1 f W B > 1 2 B <1 From Cossu et al, submitted

48 How does the presence of a canyon/ridge affect the receiving basin water properties? Mixing? Shallow level part Dense source Straight slope Exchangeable section Deep level part

49 Dense source

50 Time development of basin stratification: How does the topography-induced mixing affect the basin stratification? Salinity probes moving up and down

51 No ridge or canyon (laminar regime) C4 Distance above bottom Time Basin salinity C5

52 Canyon Larger volume of mixtures between basin and source water Less dense bottom water

53 Ridge

54 Inside canyon: different velocity, different Ri (i.e. Fr) and Re

55 δ W << α More level interface Slower flow Thicker layer Decreased entrainment δ W >> α Steeper interface Faster flow Thinner layer Increased entrainment

56 Summary role of topography: 1. Sloping topography induces a forward motion in the dense water mass (u N ) 2. Bottom friction acts as a diffusive process in the interior, upper edge becomes horizontal and arrested, lower edge moves downward creating a thin (δ) sheet of dense water 3. Small-scale topography channels water downward (β~1 gives maximum transport) and (probably) changes the mixing properties. (β > 1 = increase, β < 1 => decrease)

57 Time development of water masses: Laminar regime Highly diluted Moderately diluted Pure source water 30-70% 70-90% >90% % of total volume Canyon/ridge Straight Time Larger volumes of diluted water masses are produced with canyon/ridge than without With canyon/ridge all water is diluted, no production of pure source water

58 Wave regime % of total volume Highly diluted Moderately diluted Pure source water 30-70% 70-90% >90% Canyon/ridge Straight No production of pure source water with any topography (more overall mixing compared to laminar regime) Larger volumes of highly diluted water masses are produced with canyon/ridge than without Smaller volumes of moderately diluted (less mixed) with canyon/ridge

59 Eddy regime Highly diluted Moderately diluted Pure source water 30-70% 70-90% >90% % of total volume Canyon/ridge Straight No production of pure source water with ridge/canyon topography In eddy regime the plume water is more homogeneous compared to wave regime (fewer density classes). Stirring instead of mixing??