Top-to-bottom Ekman layer and its implications for shallow rotating flows

(1)

Top-to-bottom Ekman layer and its implications for shallow rotating flows

Cushman-Roisin, Benoit ; Deleersnijder, Eric

DOI

10.1007/s10652-018-9611-y

Publication date

2018

Document Version

Accepted author manuscript

Published in

Environmental Fluid Mechanics

Citation (APA)

Cushman-Roisin, B., & Deleersnijder, E. (2018). Top-to-bottom Ekman layer and its implications for shallow

rotating flows. Environmental Fluid Mechanics, 19 (2019), 1105–1119.

https://doi.org/10.1007/s10652-018-9611-y

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(2)

for Shallow Rotating Flows

BenoitCushman-Roisin

Thayer S hoolof Engineering

Dartmouth College

Hanover, New Hampshire03755, USA

Eri Deleersnijder

Institute of Me hani s, Materials andCivil Engineering(IMMC)

&Earthand LifeInstitute (ELI)

Université atholique deLouvain

4 avenue Georges Lemaître

B-1348 Louvain-la-Neuve,Belgium

Delft Instituteof Applied Mathemati s(DIAM)

Delft Universityof Te hnology

Van Mourik Broekmanweg 6

2628XEDelft, TheNetherlands

June 15,2018

Submittedto Environmental FluidMe hani s

(Spe ial Shallow Flows Issue)

ABSTRACT

Theanalyti alsolutionisderivedfor rotationalfri tionalowinashallowlayerofuidinwhi hthetop

andbottomEkmanlayersjoinwithoutleavingafri tionlessinterior. Thisverti alstru ture hassigni ant

impli ations for the horizontal ow. In parti ular, for a layer of water subje ted to both a surfa e wind

stress and bottom fri tion, the vorti ity of the horizontal ow is a fun tion not only of the url of the

wind stress (the lassi al resultfor deep waterknownasEkman pumping) but also of its divergen e. The

importan eofthisdivergen etermpeaksforawaterdeptharound3timestheEkmanlayerthi kness. This

means that a url-free but non-uniform wind stress on a shallowsea or lake an, through the dual a tion

of rotationand fri tion, generate vorti ity inthewind-driven urrents.

Wealsondthattheredu tionofthree-dimensionaldynami stoatwo-dimensionalmodelismoresubtle

than one ouldhave anti ipated and needs to be approa hed with utmost are. Taking thebottom stress

as dependent solely on the depth-averaged ow, even withsome veering, is not appropriate. The bottom

stress ought to in lude a omponent proportional to thesurfa e stress,whi his negligiblefor large depths

(3)

1 Introdu tion

The ee t of fri tion in rotating ows is governed by the Ekman number, and it is well known (Pri e et

al.,1987; Cushman-Roisin&Be kers, 2011) that,for anEkman number mu hsmallerthan unity,therole

of fri tion issigni ant only inthin layers nearthetop andbottom boundaries, alledEkman layers, and

o asionallynearlateralboundaries,inso- alledStewartsonlayers(Bennetts&Ho king,1973). Awayfrom

these boundary layers, inthebodyofthedomain alledtheinterior,theee t offri tion isfeltindire tly.

Theprimary signatureisthe Ekman pumping,averti al velo itythatstarts atthebaseofthetop Ekman

layer as proportional to the url of the surfa e stress and ends at the top of the bottom Ekman layer as

proportionalto the urlofthebottom stress. Onthe

β−

plane,thesurfa e andbottomstressesmaydier, andtheEkmanvelo itymayvaryovertheverti al,butonan

f −

plane,theEkmanpumping velo itymust be uniform over the verti al, and the surfa e and bottom stresses be equal (Cushman-Roisin & Be kers,

2011).

Thesituation be omesmu hmore omplexwhenthe rotatingdomainisshallowenoughfor thetop and

bottomEkmanlayerstomerge,leavingnofri tionlessinterior. Thebehavioroftheverti alvelo ityismu h

ri her, and the on ept of Ekman pumping is no longer helpful. The bibliography of this work is limited

be ause, although the governing equations are straightforward, the analyti al solution of the linearized

problemis tediousandtends to obs ure thepropertiesof theow.

Extending Ekman's se ond paper on fri tional rotating ows (Ekman, 1923) and with the fo us on

stormsurgesin shallowseas, Welander (1957) onsidered the asewhen the depthof fri tionalinuen e

is omparable tothe water depthand arrived at the on lusion that thewind-stressdivergen e may be of

the same importan e asthe wind-stress url in governing the streamfun tion of the depth-averaged ow,

but he didnot onsider theverti al velo ityand qui klypro eeded to thetime-dependent ase ofa storm

surge. Inthe1970s, whenele troni omputers rstbe ame availableandusedforstormsurgefore asting,

they were not yet apable of handling three-dimensional grids, and a urry of work ensued to redu e the

3D equations to 2D equations bypre-solving for theverti al prole of thehorizontal velo ity. This work

(Jelesnianski, 1970; Forristall, 1974; Cheng et al., 1976; Nihoul, 1977) largely fo used on the ee ts of

a verti ally varying eddy vis osity to obtain the bottom stress in terms of the depth-averaged horizontal

velo ity. No attention waspaidto the verti al velo ity. Ade adelater, as3Dnumeri al models ameinto

being, Lyn h and O er (1985) paid renewed attention to the verti al stru ture of vis ous rotating ow

in order to provide analyti solutions astest ases for 3D hydrodynami models. Like their prede essors,

they hosedepth-varying eddyvis osities and tookthe bottom stress as aligned with and proportional to

thedepth-averaged velo ity. Theirsolutions are astassolutions to an eigenvalue problemintheverti al,

whi h requiresthenumeri al solution ofa 1Dequation.

In this work, we ask dierent questions: What happens to the on ept of Ekman pumping when the

Ekmannumberis notsmall? Whatisthe verti al proleoftheverti alvelo ity? Howdolo alwind-stress

urland divergen e ae tthedepth-averagedow? And,underwhi h onditions an thebottom stressbe

takenasparallelto the depth-averaged velo ity? Theanswerto thelatterpointsto unexpe ted di ulties

intheredu tionof3Ddynami s toa2Dmodeland might astdoubts onthea ura yofpast2Dmodels.

2 Top-to-Bottom Ekman Dynami s

We are onsidering a shallow ow ofwater ofuniform density subje tto an imposed surfa e stress at the

top andno slip alonga at horizontalbottom.

We solve the lassi al Ekman dynami s problem in the ontext of the following assumptions:

f

-plane (with

f > 0

for onvenien e), low-Rossby number (to permit the negle t of nonlinear adve tion terms and fa ilitatetheanalysis),hydrostati balan e (

∂p/∂z = −ρg

), onstant density

ρ

(no strati ation),at bottom and rigid lid (

−H ≤ z ≤ 0

,

H =

onstant), eddy vis osity model with uniform eddy vis osity

ν

,

(4)

and imposedsurfa e wind stress (

τ

x

,

τ

y

fun tionsof

x

and

y

).

Were ognize thatthenegle tof thetemporal derivativesand nonlinearadve tion termsis notjustied

inmany ir umstan es,but wenonetheless ignorethosetermsheretopla ethefo usonthe ombinedrole

of Coriolis and fri tion for es in a mathemati ally tra table fashion, with the underlying hypothesis that

themainndings ofour study will ontinue to holdat leastqualitatively inthebroader dynami alsetting

thatin ludes temporal derivatives and nonlinearadve tion terms. It also happens thatthe assumption of

a lowRossbynumber andtemporal Rossbynumberis justiable wheneverthes ales of theproblem obey

thefollowing relations:

Ek =

ν

f H

2 ≥ 1,

Ro =

U

f L

=

τ

ρf

2 HL

≪ 1,

Ro

T

=

1 f T

≪ 1,

(1)

in whi h

L

and

U

are horizontal length and velo ity s ales,

T

is the time s ale, and

τ

is a s ale for the windstress. Su hrelationsarenotunrealisti forshallowwaterows. For example,thevalues

f ∼ 10

−4

/s,

H ∼ 10

m,

L ∼ 10

km,

T ∼ 2 × 10

5

s (=several days),

τ /ρ ∼ 10

−4

m

2

/s

2

, fri tionvelo ity

u

∗

=

pτ/ρ ∼

10 −2

m/s,

U ∼ τ /ρf H ∼ 0.1

m/s,and

ν ∼ u

∗

H ∼ 10

−1

m

2

/s do meet the pre eding riteria (

Ek ∼ 10

,

Ro ∼ 0.1

,and

Ro

T

∼ 0.05

).

∂v

g

∂y

= 0.

(4)

It isalso

z

-independent byvirtue ofthe hydrostati balan e with onstant density. The equations governing thethree velo ity omponents

u

,

v

and

w

are:

−f v = −f v

g

+ ν

∂

2 u

∂z

2

(5)

+f u = +f u

g

+ ν

∂

2 v

∂z

2

(6)

∂u

∂x

+

∂v

∂y

+

∂w

∂z

= 0.

(7)

There arethree boundary onditions at ea hhorizontal boundary.

•

Atthe bottom,weimposeno ow onthefri tionaland impermeablebottom:

u = v = w = 0

at

z = −H.

(8)

•

Atthe top,we impose asurfa e wind stressand noverti alowthrough therigid lid:

ρν

∂u

∂z

= τ

x

,

ρν

∂v

∂z

= τ

y

,

w = 0

at

z = 0.

(9)

(5)

WenotethatthesetofEquations(5)-(6) -(7) ontains atotalofve

z

-derivativeswhilewehavespe ied six boundary onditions in the

z

-dire tion. Thus, the system is over-spe ied by one, and we ought to expe t a onstraint on some of the variables that enter the equations, parti ularly a relationship between

thegeostrophi velo ity(

u

g

,

v

g

),whi h isthehorizontal pressuregradient indisguise,and thewind stress (

τ

x

,

τ

y

).

Thesituation anbeanti ipatedby onsideringthe aseofadeepwater olumn omparedtotheEkman

layerthi kness, dened as

d =

r 2ν

f

.

(10)

If

d ≪ H

, fri tion is signi ant only in top and bottom boundary layers that o upy mu h less than the waterdepth,leavingafri tionlessandgeostrophi interior. Inthis ase, surfa eEkmandynami sgenerates

aso- alled Ekmanpumping verti alvelo itythroughthe interior(Cushman-RoisinandBe kers, page254)

proportional tothe wind stress url:

w

interior

=

1 ρf

∂τ

y

∂x

−

∂τ

x

∂y

,

(11)

while bottom Ekman dynami s generate an interior verti al velo ity proportional to the vorti ity of the

geostrophi ow(Cushman-Roisin andBe kers, page 249):

w

interior

=

d

2 ∂v

g

∂x

−

∂u

g

∂y

.

(12)

Given that the verti al velo ity in the interior takes a unique value be ause theinterior geostrophi ow

hasno divergen e, itfollowsthat the pre edingexpressions mustbeequal to ea h other:

∂v

g

∂x

−

∂u

g

∂y

=

2 ρf d

∂τ

y

∂x

−

∂τ

x

∂y

.

(13)

Inotherwords,inwaterdeep omparedtotheEkmanlayerthi kness,thevorti ityoftheinteriorgeostrophi

owis onstrainedtobeproportionaltothe urlofthesurfa ewindstress,alo aldependen y. Theprimary

aim of this study is to determine how this relationship is ae ted when thewater depth is omparable to

theEkman layerthi kness(

d ∼ H

,i.e.

ν/f H

2 ∼ 1

)and fri tion ae tstheentire water olumn.

At this point, we ould pro eed to solve Equations (5) -(6)for

u

and

v

,but the algebra is very tedious (Welander, 1957). A more e ient path toward the relationship we seek is not to solve for the velo ity

omponentsthemselvesbut fortheir divergen e

∂u/∂x + ∂v/∂y

andvorti ity

∂v/∂x − ∂u/∂y

. Forthis,we dierentiate Equations (5)-(6) and ombine to obtain the following oupled equations for the divergen e

and vorti ity:

f

∂u

∂x

+

∂v

∂y

= f

∂u

g

∂x

+

∂v

g

∂y

+ ν

∂

2 ∂z

2 ∂v

∂x

−

∂u

∂y

(14)

f

∂v

∂x

−

∂u

∂y

= f

∂v

g

∂x

−

∂u

g

∂y

− ν

∂

2 ∂z

2 ∂u

∂x

+

∂v

∂y

.

(15)

In (14),thedivergen e ofthegeostrophi owvanishesbyvirtue of (4) , and thegeneral solution ofthe

equations an be written as:

∂u

∂x

+

∂v

∂y

= cosh ζ (D cos ζ − C sin ζ) + sinh ζ (B cos ζ − A sin ζ)

(16)

∂v

∂x

−

∂u

∂y

=

∂v

g

∂x

−

∂u

g

(6)

inwhi h thedimensionlessverti al oordinate

ζ

isdened by

ζ =

z + H

d

.

(18)

It vanishes at thebottom and rea hes

H/d

at thesurfa e.

The four onstants ofintegration

A

,

B

,

C

and

D

are determined bythe appli ation oftop and bottom boundary onditions. Thehorizontalvelo ityvanishesalongthebottom(

ζ = 0

),andsomustitsdivergen e and vorti ity. Thisdeterminesthe oe ients

A

and

D

:

A = −

∂v

g

∂x

−

∂u

g

∂y

,

D = 0.

(19)

Before applying boundary onditions at the surfa e, it is ne essary to determine the verti al velo ity

a rossthewater olumn,whi h isobtained byintegrating the ontinuity equationverti ally,starting from

w = 0

at thebottom:

w = −

Z

z

0 ∂u

∂x

+

∂v

∂y

dz

′

= −d

Z

ζ

0 −C cosh ζ

′

_{sin ζ}

′

_{+ sinh ζ}

′

_{(B cos ζ}

′

_{− A sin ζ}

′

₎

dζ

′

=

d

2 (B + C)(1 − cosh ζ cos ζ) −

d

2 (B − C) sinh ζ sin ζ

+

d

2 A (cosh ζ sin ζ − sinh ζ cos ζ).

(20)

Enfor ing

w = 0

atthe surfa e (

z = 0

,

ζ = H/d

)yields:

A

cosh

H

d

sin

H

d

− sinh

H

d

cos

H

d

+ (B + C)

1 − cosh

H

d

cos

H

d

− (B − C) sinh

H

d

sin

H

d

= 0 .

(21) Theappli ationoftheremainingtwoboundary onditionsatthesurfa e,whi hinvolvethesurfa ewind

stress,ne essitates the preliminarytaking ofthe

z

-derivative of thedivergen e andvorti ity:

∂

∂z

∂u

∂x

+

∂v

∂y

=

1 d

[(B − C) cosh ζ cos ζ − (B + C) sinh ζ sin ζ]

−

1 d

A(cosh ζ sin ζ + sinh ζ cos ζ)

(22)

∂

∂z

∂v

∂x

−

∂u

∂y

=

1 d

[(B + C) cosh ζ cos ζ + (B − C) sinh ζ sin ζ]

−

1 d

A(cosh ζ sin ζ − sinh ζ cos ζ) .

(23)

Thestress boundary onditions (9) an nowbe applied at thetop (

ζ = H/d

):

∂τ

x

∂x

+

∂τ

y

∂y

=

ρf d

2 (B − C) cosh

H

d

cos

H

d

− (B + C) sinh

H

d

sin

H

d

−

ρf d

2 A

cosh

H

d

sin

H

d

+ sinh

H

d

cos

H

d

(24)

∂τ

y

∂x

−

∂τ

x

∂y

=

ρf d

2 (B + C) cosh

H

d

cos

H

d

+ (B − C) sinh

H

d

sin

H

d

−

ρf d

2 A

cosh

H

d

sin

H

d

− sinh

H

d

cos

H

d

.

(25)

(7)

Sin e the oe ient

A

is already known in terms of the vorti ity of the geostrophi ow [see (19) ℄, the pre edingtwo equations forma 2

×

2systemof equations for

B − C

and

B + C

. The solutionis:

B − C =

2 ρf d∆

∂τ

y

∂x

−

∂τ

x

∂y

sinh

H

d

sin

H

d

+

2 ρf d∆

∂τ

x

∂x

+

∂τ

y

∂y

cosh

H

d

cos

H

d

−

1 ∆

∂v

g

∂x

−

∂u

g

∂y

cosh

H

d

sinh

H

d

+ cos

H

d

sin

H

d

(26)

B + C =

2 ρf d∆

∂τ

y

∂x

−

∂τ

x

∂y

cosh

H

d

cos

H

d

−

2 ρf d∆

∂τ

x

∂x

+

∂τ

y

∂y

sinh

H

d

sin

H

d

+

1 ∆

∂v

g

∂x

−

∂u

g

∂y

cosh

H

d

sinh

H

d

− cos

H

d

sin

H

d

,

(27)

after

A

wasrepla ed byits valuegivenin(19) andwith

∆

,thedeterminant ofthesystem, dened as

∆ = cosh

2 H

d

cos

2 H

d

+ sinh

2 H

d

sin

2 H

d

= sinh

2 H

d

+ cos

2 H

d

.

(28)

Substitutionin(21)yieldsthesought-afterrelationshipbetweenthegeostrophi owandthewindstress:

∂v

g

∂x

−

∂u

g

∂y

cosh

H

d

sinh

H

d

− cos

H

d

sin

H

d

=

2 ρf d

∂τ

y

∂x

−

∂τ

x

∂y

cosh

2 H

d

+ cos

2 H

d

− cosh

H

d

cos

H

d

− 1

+

2 ρf d

∂τ

x

∂x

+

∂τ

y

∂y

sinh

H

d

sin

H

d

.

(29)

This last equation is the relation between the geostrophi ow (

u

g

,

v

g

), and the wind stress (

τ

x

,

τ

y

) that extends Equation(13) to the ase of shallowwater(

d ∼ H

,i.e.

ν/f H

2 ∼ 1

). Itwasderived byWelander (1957, Equation (17)) albeit asa more ompli ated mathemati al expression. We note that thevorti ity

of the geostrophi ow is no longer simply proportional to the url of the wind stress but is now a linear

ombination of both url and divergen e of thewind stress. In thelimit

H ≫ d

,Equation (29) reverts to (13)asit should.

To better dis ern the dependen y of the geostrophi vorti ity on the url and divergen e of the wind

stress,we rewrite (29) as:

∂v

g

∂x

−

∂u

g

∂y

=

2 ρf d

F

H

d

∂τ

y

∂x

−

∂τ

x

∂y

+

2 ρf H

G

H

d

∂τ

x

∂x

+

∂τ

y

∂y

,

(30)

equalto thedepth-averagedow,but isno longersointhe aseofshallowwater be auseEkmandynami s

a tovertheentire water olumn. Thus,weneed tomake adistin tionbetween thegeostrophi omponent

of theowand the depth-averaged ow.

Wedene thedepth-averaged velo ityas

¯

u =

1 H

Z

0 −H

u dz ,

¯

v =

1 H

Z

0 −H

v dz .

(31)

Verti alintegration of Equations(14) -(15) fromtop to bottom

†

followed byutilization ofboundary

ondi-tions (9)and useof (22)-(23)provides:

∂ ¯

u

∂x

+

∂¯

v

∂y

=

1 ρf H

∂τ

y

∂x

−

∂τ

x

∂y

−

d

2H

(B + C)

(32)

∂¯

v

∂x

−

∂ ¯

u

∂y

=

∂v

g

∂x

−

∂u

g

∂y

−

1 ρf H

∂τ

x

∂x

+

∂τ

y

∂y

+

d

2H

(B − C).

(33)

Further repla ement of

B + C

and

B − C

bytheirvalues givenin(27) -(26) leads to:

∂ ¯

u

∂x

+

∂¯

v

∂y

= −

d

2H∆

∂v

g

∂x

−

∂u

g

∂y

cosh

H

d

sinh

H

d

− cos

H

d

sin

H

d

+

1 ρf H

∂τ

y

∂x

−

∂τ

x

∂y

1 −

1 ∆

cosh

H

d

cos

H

d

+

1 ρf H

∂τ

x

∂x

+

∂τ

y

∂y

1 ∆

sinh

H

d

sin

H

d

(34)

∂¯

v

∂x

−

∂ ¯

u

∂y

=

∂v

g

∂x

−

∂u

g

∂y

1 −

d

2H∆

cosh

H

d

sinh

H

d

+ cos

H

d

sin

H

d

+

1 ρf H

∂τ

y

∂x

−

∂τ

x

∂y

1 ∆

sinh

H

d

sin

H

d

−

1 ρf H

∂τ

x

∂x

+

∂τ

y

∂y

1 −

1 ∆

cosh

H

d

cos

H

d

.

(35)

†

(9)

Finally, eliminationof the vorti ityof the geostrophi ow byuseof (29)provides:

∂ ¯

u

∂x

+

∂¯

v

∂y

= 0,

(36) and

∂¯

v

∂x

−

∂ ¯

u

∂y

=

2 ρf d

F

¯

H

d

∂τ

y

∂x

−

∂τ

x

∂y

−

2 ρf d

G

¯

H

d

∂τ

x

∂x

+

∂τ

y

∂y

.

(37)

The rst of these equations returns the expe ted result that the depth-averaged ow has no divergen e

(in ompressible owbetween two parallel impermeableboundaries), while these ond equationis the

on-strainton theow[Equation (29) ℄nowexpressedintermsofthevorti ityofthedepth-averagedow. This

equation was rstderived by Welander (1957, Equation (26)), but no dis ussion wasprovided otherthan

pointingout thatthere isa dependen y onthewind-stressdivergen e.

Figure 2: The dimensionless fun tions

¯

F

and

¯

G

of thethi kness ratio

H/d

. The fun tion

¯

F

provides the dependen eof the depth-averaged vorti ityon the wind-stress url,and

G

¯

on thewind-stressdivergen e.

Figure2showsthedependen yofthe

F

¯

and

G

¯

fun tionsonthethi knessratio

H/d

. Thedependen yof thedepth-averaged vorti ityonthewind-stress url(

F

¯

fun tion) isfairlysimilartothatofthegeostrophi vorti ity(

F

fun tion),startingfrom

0

andasymptotingto

1

;theintermediatemaximumnowis

0.9259

,and there isan intermediate minimum of

0.8903

. In ontrast, thedependen y on thewind-stressdivergen e is quitedierent. First,thereisareversalinsign(notshowninFigure2but madeexpli itinEquation(37) ).

Indeep water (

H/d → ∞

),itslowly vanishes;intheshallowlimit (

H/d → 0

),thefun tion

G

¯

rea heszero morerapidlythanthefun tion

¯

F

,implyingthat,when

H << d

,thewind-stress urlremainsthedominant ontribution to the depth-averaged vorti ity (as shown in the next se tion). The fun tion

G

¯

rea hes a maximum of

0.1730

for

H/d = 3.17

. In other words, the ontribution of the wind-stress divergen e to the depth-averaged vorti ity is only signi ant at intermediate depths. Figure 3 displays the ratio of the

oe ients of the divergen e and url terms in (37) (without the minus sign). We note that it rea hes a

maximumof

0.1891

(19%) at

H/d = 3.265

.

3 Case of Very Shallow Flow

(10)

Figure3: The ratio ofthe oe ientsof the divergen e and url termsinEquation(37) .

thelimit

H/d → 0

inboth (29)and (35) . At the leading order, these equations be ome, respe tively:

∂v

g

∂x

−

∂u

g

∂y

≃

3 2ρf H

∂τ

x

∂x

+

∂τ

y

∂y

(38)

∂¯

v

∂x

−

∂ ¯

u

∂y

≃

H

ρf d

2 ∂τ

y

∂x

−

∂τ

x

∂y

.

(39)

Wenotefrom Equation(38) thatthevorti ityofthe geostrophi owis now proportionalto the

diver-gen eofthewind stress,a ompleteipfrom(13)where itwasproportionalto the urlofthewindstress.

Equation(39) forthe depth-averaged ow issimilar to(13) but witha oe ient thathasan extrapower

of theratio

H/d

.

4 Impli ations for 2D Modeling

(42)

inwhi h thewindstress isapplied asabodyfor eand

r

isafri tion oe ient,whi hshouldberelatedto thevis osity

ν

somehow. We useoverbars onthevelo ity omponentsto indi atethat theseareverti ally uniform, i.e. theyaredepth averages. Thus, this modelassumes a bottom stress ounter-alignedwiththe

depth-averagedvelo ity. Su ha 2Dmodel,oftenextendedto in lude timederivatives, nonlinearadve tion

terms, and aquadratri expressionfor the bottom stress haslongbeen astaple ofwind-driven ir ulation

modeling for lakes and oastal areas (Heaps, 1969 - Equation (5); Li k, 1976 - Page 58; S hwab, 1983

(11)

-We ask what value should be given to the fri tion oe ient

r

in terms of the kinemati vis osity

ν

and the other parameters in the formalism,

f

and

H

, in su h a wayas to render this 2Dmodelan exa t representation ofthe 3Dmodelaftereliminationofthe verti al stru tureoftheow. For this,webeginby

solving themomentumequations (40) -(41) for

u

¯

and

v

¯

,whi h isstraightforward:

¯

u =

f

2 f

2 _{+ r}

2 u

g

−

rf

f

2 _{+ r}

2 v

g

+

1 ρH(f

2 _{+ r}

2 ₎

(rτ

x

+ f τ

y

)

(43)

¯

v =

f

2 f

2 + r

2 v

g

+

rf

f

2 + r

2 u

g

+

1 ρH(f

2 + r

2 )

(rτ

y

− f τ

x

) ,

(44) andthenpro eedbytakingthedivergen e

∂ ¯

u/∂x+∂¯

v/∂y

,whi hwesettozeroas ontinuity(42)demands. Theresult is:

∂v

g

∂x

−

∂u

g

∂y

=

1 ρrH

∂τ

y

∂x

−

∂τ

x

∂y

+

1 ρf H

∂τ

x

∂x

+

∂τ

y

∂y

,

(45)

whi h is isomorphi to (30) in that the vorti ity of the geostrophi ow is a linear ombination of the

divergen e and url ofthe wind stress. Mapping(45) onto (29) demands thefollowing two equivalen ies:

r

f

=

d

2H

cosh

H

d

sinh

H

d

− cos

H

d

sin

H

d

cosh

2 _H

d

+ cos

2 H

d

− cosh

H

d

cos

H

d

− 1

(46)

2H

d

sinh

H

d

sin

H

d

= cosh

H

d

sinh

H

d

− cos

H

d

sin

H

d

.

(47)

Whiletherstoftheseequationsservestosetthevalueforthefri tion oe ient

r

intermsofthevis osity

ν

(subsumedin

d

),the se ondequationought to be satised,too. A glan eatthis equationrevealsthatit is impossible to satisfyit for an arbitraryvalue

†

of theratio

H/d

. Thus, we arebrought to on lude that the redu ed two-dimensional model (40) -(41) -(42) is not a legitimate redu tion of the three-dimensional

model(5)-(6) -(7) .

One may think at this point that the veering involved in Ekman dynami s should ause the bottom

stress tobe ata non-zero angleto themean velo ityandthat thepresen eofthis angle wouldresolvethe

matter. Surprisingly, however, it does not: Equation (47) arises on eagain. A non-intuitive remedy is to

turnthe surfa ewindstressbya ertainangle. Thisdoesnot seemtohave aphysi albasisuntilonethinks

of it as a parameterization of the bottom stress as a linear ombination of the mean velo ityand surfa e

stress ratherthan beingdependent onthe meanvelo ityalone.

For an angle

α

of rotation, positive inthetrigonometri sense, we try2Dequations of theform:

−f ¯

v = −f v

g

+

cos α τ

x

− sin α τ

y

ρH

− r¯

u

(48)

+f ¯

u = +f u

g

+

sin α τ

x

+ cos α τ

y

ρH

− r¯

v

(49)

∂ ¯

u

∂x

+

∂¯

v

∂y

= 0.

(50)

of whi h thesolution for

u

¯

and

v

¯

is:

¯

u =

f

2 u

g

− rf v

g

f

2 _{+ r}

2 +

(f sin α + r cos α)τ

x

+ (f cos α − r sin α)τ

y

ρH(f

2 _{+ r}

2 ₎

(51)

¯

v =

f

2 v

g

+ rf u

g

f

2 _{+ r}

2 +

(f sin α + r cos α)τ

y

− (f cos α − r sin α)τ

x

ρH(f

2 _{+ r}

2 ₎

,

(52)

†

(12)

withthe following zero-divergen e ondition:

∂v

g

∂x

−

∂u

g

∂y

=

f cos α − r sin α

ρrf H

∂τ

y

∂x

−

∂τ

x

∂y

+

f sin α + r cos α

ρrf H

∂τ

x

∂x

+

∂τ

y

∂y

.

(53)

Mapping(53) onto (29) demands thefollowing two equivalen ies:

f

r

cos α − sin α =

2H

d

cosh

2 _H

d

+ cos

2 H

d

− cosh

H

d

cos

H

d

− 1

cosh

H

d

sinh

H

d

− cos

H

d

sin

H

d

(54)

f

r

sin α + cos α =

2H

d

sinh

H

_d

sin

H

_d

cosh

H

_d

sinh

H

_d

− cos

H

_d

sin

H

_d

.

(55)

Both onditions an now be met as the two onditions now form a

2 × 2

systemof equations for the two unknowns

r/f

and

α

. Figure 4 displays the variations of the ratio

r/f

(magnied by a fa tor

100

) and angle

α

as fun tions of the thi kness ratio

H/d

. As an be expe ted, both vanish for great depths (with

r/f ≃ d/2H

). As the waterdepthde reases, fri tionbe omesmore important, asexpe ted, and theangle

α

swings between a minimum of

−8.910

◦

and a maximum of

+48.17

◦

. The angle

α

be omes appre iable (

|α| > 5

◦

) when

H/d

fallsbelow

6

.

Figure 4: The fri tion fa tor

r

(s aled and magnied) and angle

α

of veering (in degrees) that are the solutions to Equations (54) -(55) .

Nowthatwehaveidentiedana eptableredu tion, we anextra tthe orrespondingexpressionofthe

bottomstress. WritingEquations (48) -(49)as

−f ¯

v = −f v

g

+

τ

x

ρH

−

τ

bx

ρH

(56)

+f ¯

u = +f u

g

+

τ

y

ρH

−

τ

by

ρH

(57)

to serve asthedenitionsof the bottomstress omponents (

τ

bx

, τ

by

),we obtain:

τ

bx

= (1 − cos α) τ

x

+ sin α τ

y

+ ρrH ¯

u

(58)

τ

by

= (1 − cos α) τ

y

− sin α τ

x

+ ρrH ¯

v .

(59) Thus, theproperwayto parameterizethebottom stressis toadd asurfa e stress omponent toa lassi al

(13)

forin reasing depthbe ausetheangle

α

vanishesas

H/d → ∞

,asFigure4shows. Theneedtoin ludethe surfa e stress inthe formulation of the bottom stress was already re ognized byNihoul (1977 - Equation

(61)), but without re ourse to the pre eding developments, he did not make provision for veering and

obtained a oe ient (

m

inhis notation)witha weak andmonotoni dependen e on thewaterdepth.

5 Con lusions

We investigated the ase of rotating shallow water when Ekman dynami s extend throughout the water

olumn insteadofbeingrestri tedtothinboundarylayers undersurfa e andabove bottom. Although this

had been investigated in parts some time ago, the omplexity of the algebra obs ured some of the ow

properties. Chiey,Welander (1957) did identify that thedepth-averaged ow vorti ity dependsnot only

onthesurfa estress url(asitdoesindeepwater)butalsoonitsdivergen e;however,hedidnotdo ument

the importan e of the new term. Using less ompli ated algebrai manipulations, we show here that the

divergen e term vanishes in both limits of very deep (

H/d → ∞

) and very shallow water (

H/d → 0

), withlargest signi an e on the orderof 20% for awaterdepth around3 times theEkman layer thi kness

d =

p2ν/f

.

Thephysi alinterpretationofthedependen eoftheverti alvelo ity,andhen eoftheowvorti ity,on

both urlanddivergen eofthesurfa estressisasfollows. Inredu eddepth,wherethetopandbottom

Ek-manlayersoverlap,ea hoftheselayers

†

is, inasense,in omplete. Thisimpliesthattheverti allyaveraged

transportinthetoplayerisatlessthanthe full

90 ◦

to the surfa estress. The divergen e ofthistransport,

whi hindu estheverti alvelo ity,thus ontainstwoterms: Thetransport omponentperpendi ulartothe

surfa e stress ontributes the lassi al term proportional to the url, while the omponent parallel to the

surfa estress ( ausedbythedeparture from

90 ◦

) ontributes ase ondterm proportionaltothedivergen e

of thesurfa e stress. Thelatter ontribution vanishes whenthe depthbe omessu ient large to reate a

de oupling of both Ekman layers, with the transportin the top Ekamn layer then oriented at

90 ◦

to the

surfa e stress.

Although the surfa e stress divergen e ee t annot be dominant in the presen e of a nite surfa e

stress url, it will dominate wherever and whenever the surfa e stress url is weak or nil in water with

depth omparable to the Ekman layerthi kness. Thismeans thata url-free but non-uniform wind stress

on a shallow sea or lake an, through the dual a tion of rotation and fri tion, generate vorti ity in the

wind-driven urrents. To the knowledgeof theauthors,this had not been identied earlier.

This analysis also has impli ations for the redu tion of three-dimensional hydrodynami s to a

two-dimensional model. We show that, for su h a redu tion to be faithful to 3Ddynami s, itis inappropriate

to alignthebottom stresswiththe depth-averaged velo ity,and this astdoubts onnumeroussimulations

performedoverthelastvede adeswithtwo-dimensionalmodels. Wefurthershowthatinvoking aveering

anglebetween bottom stress anddepth-averagedowstill doesnot uretheproblem. Instead,thebottom

stress must be taken as onsistingof two omponents, one depending on the surfa e stress (with veering)

and theotherdependingon the depth-averaged ow(without veering).

ACKNOWLEDGEMENTS

The rst author expresses his gratitude to Prof. GertJan van Heijst for having organized the 2017

Symposiumon Shallow Flowsin Eindhoven. These ond authorwishes to a knowledge past supportfrom

theBelgianFund forS ienti Resear h(F.R.S.

−

FNRS)inre ognition ofthefa tthatsome elementsof thepresent paperoriginated while heserved asResear h Asso iateof theFNRSearlier inhis areer.

†

Thereissomesubje tivityastowherethetoplayerendsandthebottomonebegins,obviously,butthisdoesnotneedto

(14)

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