UNIVERSITY OF GOTHENBURG Department of Earth Sciences Geovetarcentrum/Earth Science Centre

ISSN 1400-3821 B 614 Bachelor of Science thesis Göteborg 2010

Mailing address Address Telephone Telefax Geovetarcentrum Geovetarcentrum Geovetarcentrum 031-786 19 56 031-786 19 86 Göteborg University S 405 30 Göteborg Guldhedsgatan 5A S-405 30 Göteborg SWEDEN

A new entrainment parametrisation for rotating

bottom gravity currents

Magnus Nilsson

Abstract

This paper has been written to introduce a new entrainment parametri-sation for rotating bottom gravity currents. The parametrisation isbased on rotating plume dynamics first described in Umlauf & Arneborg(2009 a,b), two articles describing the dynamics of a buoyancy drivenflow in the Western Baltic Sea. The general idea is that entrainmentcan be parametrised as an effect of interfacial dynamics, more preciselyas dependent on the effect of transverse transport on the interfacethickness and therefore on the gradient Richardson numbers and over-all stability of the interfaces. Results from use of the new entrainmentparametrisation is then compared to results from existing parametri-sations founded both on laboratory work, data-fitting from oceanicconditions and more theoretical approaches, as well as to observed en-trainment rates. Comparisons suggest that the new parametrisation isan improvement on all the other parametrisations in this study whenrotational dynamics are guiding the flow. As an assessment of the va-lidity of the assumed dynamics the parametrisation is dependent onthe Rossby number, Ro, where a small Rossby number implies thatrotational effects are important. Included is also a calibration of theentrainment formula for an empirical constant γ which results in real-istic values of all to γ related parameters.

1 Introduction

Recent theoretical and observationalwork on rotating bottom gravity cur-rents has unveiled a number of im-portant mechanistic features. Herepresented is an attempt to use find-ings from Umlauf & Arneborg (2009a,b), especially the described wedgeshaped interfacial layer and thegeostrophically balanced interfacialjet, to produce or rather to finetune a more mechanistic entrainmentparametrization first introduced inNilsson (2010) for such currents withfocus kept on interfacial dynamics.

Most entrainment parametrisationsin use today are based on laboratorywork and the statistical nature of en-training fluid flows. This report willhopefully add a more process based ormechanistic view of the phenomenon.

In this section there will be abrief introduction to the differentparametrizations on which the com-parative study is based. First out isthe laboratory work based and stillwidely used Ellison & Turner (1959)model. That particular parametriza-

1

tion, being dependent on the bulkFroude number, suffers from havinga lower bound on the set of Froudenumbers that induces mixing, i.e. acritical Froude numbers Fr = 1.25.However, in many gravity currentsthroughout the world the bulk Froudenumbers are often subcritical and yetthe entrainment of ambient water issubstantial. In addition entrainmentestimates using the Ellison Turner(1959) model on flows featuring su-percritical Froude numbers are oftenexaggerations in comparison to ob-servations for large scale flows as weshall see in section three. The secondparametrisation is an improved bulkFroude number based entrainmentformula presented in Cenedese et al.(2004), that is also based on labora-tory work. It has the advantage thatentrainment is induced even for sub-critical flows and it is also smootherin the sense that it does not exagger-ate entrainment rates for high Froudenumber flows as much as the Elli-son Turner model is prone to. Thethird candidate is a more mechanis-tic approach suggested in Stigebrandt(1989), that is an extension of theKato & Phillips entrainment formula,however, derived for a bottom grav-ity current. In general the entrain-ment estimates given by this modelare quite well in line with observedvalues. However, in the derivationof this formula it is assumed that ahomogeneous bottom current is sep-

arated from the ambient water bya sharp density interface, a generaldescription that is not in agreementwith observations on the locations ofthis study, as these oceanic gravitycurrent are best described as havinga nearly homogeneous bottom layerthat is separated from the ambientwater by a stratified interfacial layerwith a thickness comparable to thatof the bottom layer. This discrep-ancy with regard to current descrip-tions tends to make the validity of thederivation that leads to the model inStigebrandt (1989) questionable formany of the flows in this study. In ad-dition there is a forth parametrisationpresented in Arneborg et al. (2007)for flows with low froude numbersand weak slopes. That particularparametrization is not only Froudebut also Ekman number dependent,which could prove important in shal-lower and more friction dependentflows. The fifth and final parametri-sation in this comparison is from arecent paper by Cenedese & Adduce(2010). It is a formula based on data-fitting from a vast amount of oceanicas well laboratory data. It is bothFroude and Reynoldes number de-pendent for the laboratory flow, how-ever, for the oceanic flows a Reynoldsnumber of 107 is assumed and hencein this comparison it is only Froudenumber dependent.

We will compare the results using

2

the new entrainment formula as wellas the five others by Ellison Turner,Cenedese, Stigebrant, Arneborg andCenedese & Adduce respectively toobservations at a number of locations,seen in figure 1. The gravity currentson which this analysis is based andsome publications describing themare the following. The Red Sea out-flow (Peters & Johns 2005, a,b), theDenmark Strait overflow(Girton &Sanford 2003), the Faroe Bank chan-nel owerflow (Fer et al. 2010 & Mau-ritzen et al. 2005), the Mediterranean

outflow (Baringer & Price 1997 a,b),the Storfjorden overflow (Fer et al.2003), the outflow of dense shelf wa-ter from the Drygalski Trough in thenorthwestern Ross Sea (Munch et al.2008) and the Arkona Basin inflow(Arneborg et al. 2007). This is a setof locations that features a vast geo-graphic spread and bottom currentsconfined to channels as well as non-channel flows, a great variety in thedensity differences between ambientwaters and the bottom currents aswell as varying bottom topography.

Figure 1: Bottom current locations (a)Red Sea North Channel, (b)DanishStrait Overflow, (c)Faroe Bank Channel, (d)Arkona Basin Inflow,(e)Mediterranean Outflow, (f)Ross Sea Drygalski Trough, (g)StorfjordenOverflow

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2 Theory

Derivations in this section is donewith simple model bottom gravitycurrent in mind. Such a current inhere consists of a nearly homogeneousbottom layer that is kept homoge-neous by bottom stress induced tur-bulence. On top of the bottom layerthere is a stratified interfacial layerand the stratification efficiently hin-ders the spread of bottom originatingturbulence to the interfacial layer. Inthe bottom currents of this study theinterfacial and the bottom layer isoften of similar thickness. However,the current’s distribution between thetwo layers can be a function of thedown-stream coordinate

The downstream flow in the bot-tom and in the interfacial layers ofthe model bottom current bottomgravity current is in geostrophic bal-ance. This is because down-slopeflow under the act of rotation inducesa transverse pressure gradient man-ifested as tilting isopycnal surfaces,that in turn can drive a geostrophi-cally balanced downstream flow. Inthis model bottom current the trans-verse geostrophic transport that isinduced by the along-flow slope iscancelled out by an equally sized Ek-man transport of opposite sign, this isa necessary consequence if we assumesteady state and no along-streamvariations. However, as pointed out

in Umlauf and Arneborg (2009 b)the transport cancellation does implythe transverse velocity is zero at allz. Instead the transverse flow in thestratified upper layer is expected tobe mainly geostrophic. Given thatthe thickness of the interfacial layeris non negligible in comparison to thetotal plume thickness and if the en-trainment is weak in comparison tothe bottom friction. The nullificationof transverse transport is achievedthrough a frictional counter currentin the homogeneous bottom layer.This means that the interfacial layermainly flows along lines of constant z,as seen in figure 2, giving a flow com-ponent qi to the right of the bottomcurrent in the Northern Hemisphere.This is the so called interfacial jet,observed and described in Umlauf &Arneborg (2009 a,b).

x

y

q

qi

Figure 2: The predominant circula-tion in the interfacial layer, the diag-onal lines are indicating constant z.Here q is the total interface flow andqi its y-component

4

qi

we we

we we

W

z

y

x

Figure 3: Schematic transverse circulation

Figure 3 shows the model rotatingbottom gravity current flowing downa slope, with the positive x-axis ori-ented in the direction of the flow.All along-stream variations, that isin the x-direction, are assumed smalland our bottom current is describedby a two layer model, consisting ofa stratified interfacial layer on top ofa homogeneous bottom layer. Thegoal is to arrive at an expression forbulk entrainment by use of interfa-cial dynamics. I.e. to link the en-trainment rate to the geostrophicallybalanced transverse flow in the in-terfacial layer, the interfacial jet qi.The jet’s control of the entrainmentrate is achieved through its transversetransports influence on the interfacethickness, Hi. The interfacial jet assketched in figure 3 transports wa-ter from the right to the left side ofthe interface which lowers the gradi-ent Richardson numbers on the rightside and thus induces turbulent mix-

ing on that same side, this is the keymechanism in this parametrisation.The gradient Richardson number isdefined as

Ri =g′bHi

U2b

, (1)

where the v component of the shearis assumed small in comparison to theu component and is thus neglected .Generally, subscripts i, b, a refers to avariable for the interfacial, the bot-tom and the ambient water layer re-spectively. So that Hi is the thicknessof the interfacial layer, Ub is the veloc-ity of the bottom layer of the plumerelative to the ambient water and g′bis the reduced gravity of the bottomlayer defined as

g′b = gρb − ρa

ρ0, (2)

where g is the acceleration of grav-ity, ρb is the density of the bottomlayer, ρa is the density of the ambientwater and ρ0 is a constant reference

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density. Also assumed is that the gra-dient Richardson number has a lowerbound, a critical Richardson numberRic. So that when due to transversetransport the thickness of the interfa-cial layer decreases on the right sidein figure 3 and Ri as a consequencetends to subcritical values, mixing ofambient and bottom water into theinterfacial layer becomes great, thusmaking Hi bounded on the lower endby Ric.

Hi ≥RicU

2b

g′. (3)

Given Hi’s functional dependence onRic as described in (3) and the bot-tom currents finite extension in they-direction. The dotted vertical linein figure 3 signifies the y-coordinatewhere ∂qi

∂y= 0 and hence that the

flow in the interfacial jet increasesthe interface thickness to its left anddecreases it to its right, this impliesthat the dotted boundary is situatedat a spot where Ri = Ric. If fur-ther we assume that what drives thebottom current’s entrainment of am-bient water is the Kelvin-Helmholtzinstabilities associated with subcriti-cal Richardson numbers to the rightof the dotted vertical line, a steadystate solution with no variations inthe x-direction demands that the en-trained flux of ambient water qe is di-rectly proportional to flow in the in-terfacial jet qi as seen in figure 3, with

the entrainment flux defined as

qe = kqi, (4)

hence k is to be understood as thepart of the flow qi in figure 3 that ismade up of entrained ambient water,the remainder is made up of waterentrained from the bottom layer. Amean entrainment rate we can thenbe defined as

we =qe

W, (5)

where W is the width of the cur-rent. Given the assumed geostrophicflow in the interfacial jet and negli-gible along channel variations in den-sity and layer thickness, we may de-rive an expression for vi the velocityof the interfacial jet as

vi =g′iSx

f, (6)

where Sx is the along-flow bottomslope, that is assumed to well approx-imate the along-flow slope of the topof the bottom layer on which the in-terfacial layer flows, f as the coriolisparameter and g′i is the reduced grav-ity of the interfacial layer,

g′i = gρ̄i − ρa

ρ0(7)

where,

ρ̄i =1

Hi

∫

Hi

ρidz (8)

Given that

qi = viHi, (9)

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substitution of (6) and (3) into (9)and then substitution of (9) into (4)and finally (4) into (5) yields

we = kRicU2b Sxg

′

i

fWg′b. (10)

A non-dimensional entrainment pa-rameter can then be introduced by di-viding (10) with U , the mean plumespeed, yielding

E = γRoSx, (11)

where E(

we

U

)

is the non-dimensionalentrainment parameter, Ro is the

Rossby number

Ro =U

fW(12)

and γ is a empiric constant defined as

γ = kRicg′ig′b

U2b

U2. (13)

E ′s dependence of the non-dimensional parameters Ro and Sx

will then have to be studied and anempirical γ has to be derived.

3 The different currents

This section gives a brief introductionto the different sites of interest as wellas a list of relevant parameters de-scribing the fluid flows. E.g. Froudenumber Fr, a non dimensional quan-tity that the entrainment parameterE is believed to hold a functional re-lationship to, because it is the bulkRichardson number to the power ofminus one half. Consequently suchrelationships are the basis of mostentrainment formulas. The Froudenumber is defined as

Fr =U

√

g′Hp

, (14)

where g′ is the reduced gravity of the

plume in its entirety ∆ρ = ρp − ρa.Hp is the height of the plume andU is the mean plume speed relativeto the ambient water. In additionwe present Ekman numbers Ek whichare proportional to the ratio of theEkman layer thickness to the thick-ness of the gravity current and are im-portant to the Arneborg et al. (2007)parametrization. The Ekman num-ber is defined as

Ek =UCd

fHp

, (15)

where Cd is the bottom drag coeffi-cient.

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Table 1: Bottom current caracteristicsf U ∆ρ W Hp Fr Ek Cd Sx E

RSNC 3.1 · 10−5s−1 0.56ms−1 0.2kgm−3 5km 161m 0.99 1.04 9.3 · 10−3 5 · 10−3 6.4 · 10−4

DS1 1.3 · 10−4s−1 0.7ms−1 0.34kgm−3 44km 150m 0.99 0.10 2.9 · 10−3 3.2 · 10−3 8.6 · 10−5

DS2 1.3 · 10−4s−1 1.4ms−1 0.14kgm−3 32km 240m 2.43 0.13 2.9 · 10−3 6 · 10−3 5.7 · 10−4

FBC 1.29 · 10−4s−1 1ms−1 0.36kgm−3 70km 190m 1.22 0.20 3.7 · 10−3 4.6 · 10−3 1.7 · 10−4

AB 1.19 · 10−4s−1 0.48ms−1 8.3kgm−3 12km 12m 0.49 0.67 2 · 10−3 5 · 10−4 6.6 · 10−5

MO 8.5 · 10−5s−1 0.7ms−1 0.72kgm−3 25km 110m 0.79 0.45 6 · 10−3 5 · 10−3 6.7 · 10−4

RSDT 1.3 · 10−4s−1 0.6ms−1 0.20kgm−3 15km 150m 1.10 0.08 2.5 · 10−3 5.7 · 10−2 3.3 · 10−3

SF 1.4 · 10−4s−1 0.15ms−1 0.30kgm−3 20km 35m 0.47 0.08 2.5 · 10−3 2.5 · 10−3 5 · 10−4

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The relevant parameters found in thelitterature are given in table 1. RSNCis short for Red Sea North Channel inthis section the data used is from Pe-ters & Johns (2004 a,b). The analysisis based entirely on their work in thenorth channel, as the south channelhas been left out of the analysis dueto the lack of entrainment estimates.Entrainment estimates in the northchannel are based on the method ofPhillips (1977) τe = ρUwe. wherethe entrainment stress τe has beenidentified as the interfacial stress andU as the mean plume speed. Thisbottom current is of particular in-terest to the analysis as its smallwidth and proximity to the equator(asmall Coriolis parameter) does rendera large Rossby number a conditionunder which the guiding rotationaldynamics of the new parametrizationmay be expected to break down. Inaddition, the drag coefficient, whichhas been derived from bottom stressmeasurements(τb =

√

u′w′2+ v′w′

2)

using a quadratic drag law at theheight of 3.9m above the seabed, isexceptionally large in comparison tocanonical values, which might haveimplications on the parametrisationsthat are dependent on this parameterlike Stigebrandt (1987) and Arneborget al. (2007).DS1 is short for Danish Strait sec-tion one. This section is based onthe observations of Girton & San-ford (2003). The flow in the Dan-

ish Strait features two different dy-namical regimes, where section one,defined as the section from the sillreaching about 125km in the stream-wise direction is characterized bya more moderately sloping topogra-phy and smaller entrainment rates.Entrainment estimates are acquiredfrom we = UH

ρ̄′dσ̄θ

dxwhere ρ̄′ is the

density anomaly of the entrainedwater,σ̄θ is the density of the plume,x is the downstream coordinate andU,H are plume velocity and layerthickness respectively. The Cd valuehas been found using log-fit deter-mined values of bottom stress and aquadratic drag law.DS2 is short for Danish Strait sectiontwo. This section is also based onGirton & Sanford (2003). However,it is the section farther than 125kmdownstream of the sill, that is charac-terized by a steeper topography andhigher entrainment rates. Entrain-ment and drag coefficient estimatesare acquired in the same manner asin DS1.FBC is short for Faroe Bank Chan-nel. This section is based on twoarticles, as current width estimatesfrom Mauritzen et al. (2005) are com-plemented by entrainment estimatesand bottom topograpy from Fer etal. (2010). The focus is on a sec-tion referred to as D in Fer et al.(2010), that both authors has pinpointed as a center of intensified mix-ing. The Sx value cited is the mean

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value of the tilt before and after sec-tion D. Entrainment estimates are de-rived from dissipation measurementsfollowing Arneborg et al (2007) andthe drag coefficient has been foundusing log-fit determined values of bot-tom stress and a quadratic drag law.AB is short for Arkona Basin. Thissection is based on Arneborg et al.(2007) and Umlauf & Arneborg (2009a,b). The used entrainment estimateis the 19h average from the 2007 pub-lication. This is the same channel asthe later Umlauf & Arneborg (2009a,b) is based on and thus the locationwhere the geostrophic interfacial jetand the wedge shaped interface, thatare central to the new entrainmentparametrization was first described.The location being situated in theBaltic Sea has the great virtue of be-ing virtually tide free. It features byfar the greatest density anomaly ∆ρ

and a weak slope Sx. For our Cd valuewe opted for the usage of a drag coef-ficient from the Umlauf & Arneborg(2009 a) publication, the coefficient isderived from bottom stress estimatesbased on ADV data.MO is short for Mediterranean Out-flow. The section is based onBaringer & Price (1997 a,b). Giventhe large variations in the along-flowdirection of many parameters, e.g.drag coefficient estimates varies witha factor six in the along stream direc-tion, mean values are used for all pa-rameters. Entrainment is estimated

using observed salt fluxes, Smithsmodel(Smith 1975) and volume fluxobservations. Here a mean value fromthose estimates are chosen. The dragcoefficient is given from a quadraticdrag law applied to stress approxima-tions from bulk momentum consider-ations. As a flow that is not confinedto a channel entrainment approxima-tions from this location could provean excellent measure in a deductionof whether the parametrization, thatwas derived for a channel based flowmay have an extended usage.RSDT is short for Ross Sea Drygal-ski Trough. This section is based onMuench et al. (2008). The widthstated in the table has been derivedfrom transport considerations, as nowidth is stated in the article. Thenegative sign on the Coriolis param-eter in the southern hemisphere hasbeen ignored and the Cd value iscanonical. A mean value of the en-trainment parameter E is given be-cause values varies between E ∈ [1.7 ·10−3, 6.7 ·10−3], where the higher endvalues comes from estimates of vol-ume increase, and the lower valuesfrom the same method prescribed byPeters & Johns (2005). It is also men-tioned in the article that those lowervalues are taken from stations thatare not typical of the summer of 2003excursion, and also that the mixingdata is of insufficient quality for fur-ther analysis of this discrepancy.SF is short for Storfjorden. This sec-

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tion is based on Fer et al. (2003).The value of Sx is based on a Kil-worth (2001) decent rate from thesame publication and the Cd valueis canonical. Entrainment estimatesare based on volume considerations.This flow features by far the smallestalong-stream velocity U in the analy-

sis, together with a large Coriolis pa-rameter the consequence is by far thesmallest value of the Rossby numberof all locations. This location is alsotroublesome in the sense that noneof the entrainment formulas has beenvery successful in producing adequateentrainment approximations.

4 Calibration for γ

This section is dedicated to the cali-bration of expression (11) for an ap-

propriate value of γ.

Table 2: Parameter values at the different sites. The Elit variable signifiesthe entrainment parameters we have found in the literature for the respectiveflows.

Ro Sx Elit RoSxRoSx

Elit

RSNC 3.61 0.005 6.4 · 10−4 1.8 · 10−2 28.1DS1 0.12 0.0032 8.6 · 10−5 3.8 · 10−4 4.49DS2 0.33 0.006 5.7 · 10−4 2 · 10−3 3.47FBC 0.15 0.0046 1.7 · 10−4 5.1 · 10−4 2.99AB 0.42 0.0005 6.6 · 10−5 1.7 · 10−4 2.52MO 0.33 0.005 6.7 · 10−4 1.6 · 10−3 2.44RSDT 0.26 0.057 3.3 · 10−3 1.8 · 10−2 5.29SF 0.054 0.0025 5 · 10−4 1.3 · 10−4 0.27

It is readily seen from table 2 that thevalues from RSNC and SF does not fitin with the others. In the RSNC casethis is accredited to the very largeRossby number and in a similar man-ner the SF case can be explained bythe very small value of the same pa-rameter, so both those cases can be

assumed to be dynamically dissimi-lar, although, there is no immediatereason as to why a small value of theRossby number should have a hin-dering impact on the rotational dy-namics. Nonetheless SF and RSNChave been excluded and therefore thecalibration is done using values from

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DS1, DS2, FBC, AB, MO and RSDT.The calibration is done using a leastsquares technique to minimize the er-ror e

e = |1−E

Elit| (16)

where E is the modelled entrainmentparameter and Elit is the entrainmentparameter found in the literature for

the same flow. Using (16) one arrivesat an approximate γ = 0.2602, whichis consistent with the most commonlycited value of the critical Richardsonnumber Ric = 1

4if U = Ub

2, gi =

gb2

and k = 1

2with γ as in (13) and as

seen in table 3 and in figure 4 it alsogives a good representation of data.

Ro Sx Elit E EElit

DS1 0.12 0.0032 8.6 · 10−5 1.0 · 10−4 1.17DS2 0.33 0.006 5.7 · 10−4 5.2 · 10−4 0.90FBC 0.1107 0.0046 1.7 · 10−4 1.3 · 10−4 0.78AB 0.42 0.0005 6.6 · 10−5 4.4 · 10−5 0.66MO 0.33 0.005 6.7 · 10−4 4.3 · 10−4 0.65RSDT 0.26 0.057 3.3 · 10−3 4.6 · 10−3 1.38

Table 3: Parametrized and observed entrainment using γ = 0.2602

10−4

10−3

10−2

10−1

10−5

10−4

10−3

10−2

Elit

SxRo

Figure 4: E(Ro, Sx) the locations marked as (*) are SF and RSNC

12

5 Comparison to other parametrizations

The next step is to compare resultsgiven by the different parametrisa-tions to observed entrainment. Theparametrisations included in thiscomparison is the Stigebrandt (1987)parametrisation hereafter known asES and defined by

ES = 2m0C1/2d Sx (17)

where m0 = 0.6 is an empiricalconstant, the Ellison Turner (1959)parametrisation hereafter known asEET and defined by

EET =

{

0.08Fr2−0.1Fr2+5

if Fr2 > 1.25

0 if Fr2 < 1.25

}

,

(18)the Cenedese et al. (2004) parametri-sation hereafter known as EC and de-fined by

EC = 4 · 10−4Fr3.5, (19)

the Arneborg et al. (2007) parametri-sation hereafter known as EA and de-fined by

EA = 8.4 · 10−2CdFr2.65Ek0.6 (20)

and finally the Cenedese & Adduceparametrisation hereafter known asECA and defined by

ECA =Min+ AFrα

1− CinfA(Fr0 + Fr)α(21)

with,

Cinf =1

Max+

B

Reβ(22)

where Min = 4 · 10−5, Max = 1,A = 3.4·10−3, B = 243.52, F0 = 0.51,α = 7.18, β = 0.5 and Re = 107 istaken in accordance with Cenedese &Adduce (2010).

ES

Elit

EET

Elit

EC

Elit

EA

Elit

ECA

Elit

EElit

RSNC 0.9 0 0.6 1.24 4.69 7.3DS1 2.41 0 4.5 0.70 34.90 1.17DS2 0.42 60 15.9 0.28 370.86 0.90FBC 1.97 0 4.7 1.0 70.42 0.78AB 0.40 0 1.0 0.29 0.91 0.66MO 0.69 0 0.27 0.25 0.97 0.65RSDT 1.03 0 0.17 0.02 1.85 1.38SF 0.3 0 0.06 0.01 0.11 0.07

Table 4: Parametrised entrainment normalized by observed entrainment

13

10−6

10−5

10−4

10−3

10−2

10−1

100

10−5

10−4

10−3

10−2

ES

EET

EC

EA

ECA

E

Figure 5: The drawn line is the observed entrainment and the coloured sym-bols are the entrainment parameters given by the different parametrisations

It is readily seen in table 4 and in fig-ure 5 that the difference between E

and ES, the top two candidates, intheir ability to predict entrainmentparameters is a minor one, proba-bly due to the strong dependence onthe along-flow slope present in bothparametrisations. However, if we ex-clude SF because no predictions arein line with observations at that loca-tion and RSNC because of the largeRossby number E is an improvementrelative to ES. It is also an im-provement in the sense that ES asderived in Stigebrandt (1987) is notwell defined with a thick stratified in-

terfacial layer. The Ellison Turner(1959) model has given quite bad re-sults at every attempt of usage, thiscan be attributed to the fact that anybulk Froude number Fr in our studythat exceeds 1.25 does so by rathera lot. The Cenedese parametrisationis a good improvement on the Elli-son Turner model. However, it isnot quite as accurate as E, ES orEA. The EA model does quite wellin many cases. However, it does tendto underestimate entrainment and itdoes not work well with the steeperslope of RSDT. Its underestimationof the entrainment parameter at AB

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is due to the use of a smaller dragcoefficient than in Arneborg et al.(2007). Hence, a simple recalibrationcould lead to a more realistic overallresult, as many of its predictions areon the lower end. The Cenedese &Adduce (2010) parametrisation han-dles the low Froude number flowsvery well. However, when it comesto high Froude number flows it is offby a lot. This can of course be at-tributed to the value of their Max

parameter as ECA tends to Max asFr tends to infinity and that param-eter has been chosen arbitrary. How-ever, as many of the locations are thesame in this paper as in Cenedese& Adduce (2010) it is likely that aquite different set of Froude numbershas been used. Predicting the en-trainment of SF is a failure for allmodels, perhaps with the exception ofthe ES model. However, in this caseit might be just a fortunate artefactdue to assumed guiding dynamics or abadly estimated drag coefficient. Toillustrate this potential flaw, follow-ing Stigebrandt (1987)

we =2m0u

3∗

g′H(23)

withu2

∗= U2Cd. (24)

In Stigebrandt (1987) (23) is turnedto (17) through division by U andputting

U2Cd = g′HSx (25)

obtained from the along-flow momen-tum balance. However, without theuse of (25) substitution of (24) into(23) and division by U gives the en-trainment parameter ES2

ES2 =2m0C

3/2d U2

g′H, (26)

where in (26) the assumption (25) hasnot been used, instead the fact that Uis known from data has been utilised.Using (26) yields ES2

Elit = 0.07 for theSF bottom current, which is a valuemuch more in line with the other es-timates and does suggests that per-haps the parameters describing theSF flow are erroneous or somehow in-adequate.

6 Concludings Re-

marks

As for the primary goal of this paper,to introduce and validate a new en-trainment parametrisation based ona mechanistic approach Figure 5 andtable 4 clearly demonstrates the newparametrisations predictive superior-ity over the other parametrisations.Further the calibration yields a valueof γ that is consistent with what onemight label educated guesses of thevalues of the parameters γ consistsof. Such a calibration result must belooked upon as some support favour-ing the idea that the parametrisation

15

is a representation of an actual physi-cal mechanism rather than just beingfunctional because some more diffuserelationship between the entrainmentparameter and along-flow slope or theRossby number.

For the different parametrisationsrepresented in table 5 really con-vincing results are only given by thenewly presented parametrisation andthe Stigenbrandt (1987) parametri-sation. In addition the Arneborget al. (2007) model does in somecases deliver entrainment estimatesin line with observations. However,the quality of the estimates differsquite a lot between the different flowse.g. with RSDT the estimate is offby two orders of magnitude. Noneof the parametrizations solely basedlaboratory work presented very con-vincing results. An indication thatthe entrainment phenomenon on anoceanic scale may not be duplica-ble in most laboratories, which isconsistent with the idea of a criticalReynolds number and “mixing tran-sition” e.g. in Cenedese & Adduce(2010) Rec ≈ 6 ·104 is used. Nonethe-less the Cenedese et al. (2004)parametrisation a vast improvementon the Ellison Turner (1959) modelboth in accuracy and also when it

comes to increasing the functionaldomain that is a great limitationto the Ellison Turner parametrisa-tion. The recent Cenedese & Adduce(2010) parametrisation works verywell for the low Froude number flows.For high Froude number flows, how-ever, it has the largest errors of anyparametrisation as seen in figure 5.As pointed out before this discrep-ancy is possibly due to the use of adifferent set of Froude numbers.

It seems that the new parametriza-tion even though its domain appearsto be bounded, at least on the higherend by some value of the Rossby num-ber smaller than 3.61, is more depen-dent on the along-flow slope param-eter for a good entrainment approx-imation. The question of whetherthere is a lower bound on Ro thatmust be raised as E in (11) tends tozeros as Ro tends to zero can due tothe scarcity of small Rossby numberflows not be answered using the datapresented in this paper. The along-flow slope dependence that is utilisedin both the Stigebrandt (1987) modeland the new one seems to be the mostimportant factor that distinguishesthose parametrizations as superior tothe others in table 5.

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10−4

10−2

100

10−5

10−4

10−3

10−2

Elit

Sx

10−2

100

102

10−5

10−4

10−3

10−2

Elit

Ro

100

10−5

10−4

10−3

10−2

Elit

Fr10

−210

010

210

−5

10−4

10−3

10−2

Elit

Ek

Figure 6: Ro, Sx, Fr and Ek plotted against Elit

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Figure 6 clearly shows the entrain-ment parameters dependence on thealong-flow slope, while no Rossbynumber dependence is evident. Fur-ther there is considerable scatter forboth Ekman and Froude numbers sothat the only really evident paramet-ric dependence is that on the Sx pa-rameter. A quite fortunate coinci-dence as the along-flow slope is nei-ther time dependent nor difficult tomeasure. There is of course everyreason to point out that the data onwhich our analysis is based rightfully

could be described as sparse. How-ever, the amount of locations fea-turing large scale overflows and arti-cles describing them are simply nothigh enough to support a more thor-ough investigation. There are alsothe inherit difficulties in determiningsome of the parameters in this studysuch as Cd and also E where differentmethods sometimes yield quite differ-ent results, adding to the overall un-certainties surrounding the quantita-tive nature of these phenomenon.

7 Acknowledgements

I would like to thank my supervisor Lars Arneborg for all the help andmy examiner Anna Wåhlin for valuable comments that led to significantimprovements on this bachelor thesis.

8 References

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Baringer, M. O., and J. F. Price, 1997a: Mixing and Spreading of theMediterranean Outflow. J. Phys. Oceanogr., 27, 1654–1677

Baringer, M. O., and J. F. Price, 1997b: Momentum and energy balanceof the Mediterranean outflow. J. Phys. Oceanogr., 27, 1678–1692

Cenedese, C and C. Adduce, 2010: A new parametrization for entrainmentin overflows. J. Phys. Oceanogr.,(early online release)

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Cenedese, C., J. A. Whitehead, T.A. Ascarelli, and M. Ohiwa, 2004: Adense current flowing down a sloping bottom in a rotating fluid. J. Phys.Oceanogr., 34, 188-203

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Nilsson, M., 2010: Modelling the transverse structure and entrainment inrotating bottom gravity currents, A master thesis in physical oceanography,University of Gothenburg

Peters, H., W. E. Johns, A. S. Bower, and D. M. Fratantoni 2005a: Mix-ing and entrainment in the Red Sea outflow plume. Part I: plume structure,J. Phys. Oceanogr., 35, 569–583.

Peters, H. and W. E. Johns, 2005b: Mixing and entrainment in the RedSea outflow plume. Part II: Turbulence caracteristics, J. Phys. Oceanogr.,35, 584–600.

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Smith, P. C., 1975: A streamtube model for bottom boundary currents inthe ocean. Deep-Sea Res., 22, 853–873.

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Umlauf, L. and L. Arneborg 2009a: Dynamics of a shallow gravity currentpassing through a channel. Part I: Observation of a transverse structure. J.Phys. Oceanogr., 39, 2385-2401.

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