Chapter 25

Partially Mixed Estuaries:

The Hudson River

Hartmut Peters 1, Helmut Baumert 2, and Jossy P. Jacob 1

1) Rosenstiel School of Marine and Atmospheric Science,University of Miami, Miami, USA

2) Hydromod Wissenschaftliche Beratung GbR, Wedel, Germany

25.1 Introduction

In the “classical” view of the residual circulation of estuar-ies, the “estuarine circulation” [Pritchard, 1952, Pritchard, 1954,Pritchard, 1956, Rattray and Hansen, 1962, Cameron and Pritchard, 1963,Hansen and Rattray, Jr., 1965, Pritchard, 1967], light and fresh river waterenters from the landward end, and heavy and salty ocean water enters in alower layer on the seaward side. The latter is mixed upward, and combineswith the river water to a brackish outflow toward the ocean in the upper layer.Consequently, there can not be an estuarine circulation without mixing; themixing of salty and fresh water is an integral part of the basic dynamics of thesystem. It is noteworthy that the brackish outflow is usually much strongerthan the fresh inflow. The necessity of irreversible mixing also holds in the caseof well-mixed estuaries except that the mixing might be horizontal rather thanvertical.

This chapter discusses turbulent mixing in the lower Hudson River asan example of a partially mixed estuary. Observations from 1994/95 wereanalyzed in detail in [Peters, 1997, Peters, 1999, Peters and Bokhorst, 2000,Peters and Bokhorst, 2001]. Herein, we present a summary along with excur-sions toward the turbulence modeling of the regime and an outlook towardsimilar estuaries. For our discussion, neither the classification of the Hudson aspartially mixed nor the exact definition of this term are overly important. We

1

2 CHAPTER 25. PARTIALLY MIXED ESTUARIES

rather note that, through tidal and fortnightly cycles, actual instantaneous con-ditions in the Hudson range from that of a salt wedge with pronounced two layerstructure to comparatively weak, diffuse stratification. Top-to-bottom salinitydifferences observed in 1994/95 range from about 2 to 28, and thus, given a typ-ical depth of 15 m, the stratification was never truly “weak.” Salinity herein isthe “practical salinity” [Unesco, 1981], S, a nondimensional variable. Practicalsalinity units (psu) are equivalent to parts per thousand salt concentration withhigh accuracy.

In the Hudson, as in many similar tidal estuaries, the stirring, i.e. the gener-ation of turbulence, is dominated by the tidal flow. Maximum tidal currents aretypically 1.5 m s−1, while residual currents are an order of magnitude smaller[Geyer et al., 2000]. The tidal regime of the Hudson is characterized by a pro-nounced fortnightly tidal cycle with semidiurnal amplitudes of the water levelranging from 0.45 m to 0.9 m, and by a significant monthly inequality. The riverflow varies climatologically between a maximum of 1000 m s−3 in spring anda minimum of 300 m s−3 in summer [Peters, 1999]. The estuarine flow regimecan be drastically altered on a time scale of the order of a day by events ofdownpours of the catchment basin in northern New York State and by atmo-spherically forced water level fluctuations in the adjacent ocean, in New YorkBight [Peters, 1999, Peters and Bokhorst, 2000].

This paper focuses on the “central regime” [Hansen and Rattray, Jr., 1965]of the salt intrusion in the Hudson River, where horizontal salinity gradients arefairly uniform. Our observations were taken in a reach of the river off Manhat-tan, New York City, where the channel is rather straight and where variations ofdepth and cross section are modest (Figure 25.1). During five cruises betweenMay 1994 and October 1995, each lasting from neap tides to spring tides, 26semidiurnal tidal cycles with wide range of river flow were covered by over 6,000microstructure profiles with accompanying continuous acoustic Doppler currentprofiling (ADCP) [Peters, 1999]. The microstructure profiler used carries a reg-ular SeaBird conductivity-temperature-depth (CTD) suite of sensors in additionto microstructure velocity, temperature and conductivity sensors. The primaryvariables measured by the profiler as function of depth are two pairs of electricalconductivity (C) and temperature (T ) as well as the viscous dissipation rate,ε. The ADCP yields the streamwise velocity (v, toward 30◦ true) and spanwisevelocity, u. As temperature effects on salinity (S) and density (ρ) are negligiblein the salt-stratified Hudson River we ignore T in the following. Similarly weignore u in the horizontal velocity vector, ~u = (u, v) as |u| is much smaller than|v|. The unit of ε is W kg−1 = m2s−3. An example set of microstructure profilesis depicted in Figure 25.2.

Important derived quantities include the squared buoyancy frequency, N 2 =−g ρ−1 ∂ρpot/∂z with gravity g, density ρ and potential density ρpot. Thesquared shear, V 2

z = (∂u/∂z)2 +(∂v/∂z)2, and N2 define the gradient Richard-son number, Ri = N2 V −2

z . The rms turbulent overturning scale, or Thorpescale, lth, is found from microstructure conductivity or microstructure tempera-ture data following [Thorpe, 1977]. Even though the dissipation rate ε is a basiccomponent of the energy balance of any physical system, it is not necessarily

25.1. INTRODUCTION 3

Figure 25.1: Map of the lower Hudson River Estuary with inset showing thelocations of microstructure drops. For reference, the George Washington Bridgeis at y ≈ 12 km.

the most interesting turbulence variable. Most relevant are those variables thatquantify the effect of the turbulence on the physical environment, the verticalfluxes of momentum and mass. These can be estimated from microstructureprofile data based on the turbulent kinetic eddy equation simplified by assum-ing steady state and spatial homogeneity: P + B = ε, where P is the shearproduction and B is the buoyancy flux. Within the concept of eddy viscosity(Km) and eddy diffusivity (Kρ), the two production terms become P = Km V 2

z

and B = −KρN2, respectively. Hence, when ε is measured, Km and Kρ follow

from

Km = (1 +Rf )−1

ε V −2z and (25.1)

Kρ = Rf (1 +Rf )−1

εN−2 , (25.2)

respectively, with units of m2s−1. Here, Rf = −B/P is the flux Richardsonnumber.

The procedure underlying (25.1) is known as the “dissipation method;” itoriginates from [Busch, 1977] and was first applied to oceanic observations by[Gregg et al., 1985]. It can be valid only in the absence of a significant internalwave field and provided that V 2

z is an accurate measure of all shear available togenerate turbulence. Depending on data reduction procedures, measured shearvalues can either exaggerate the true shear owing to instrumental noise or un-derestimate it because of poor vertical resolution and/or temporal averaging.

4 CHAPTER 25. PARTIALLY MIXED ESTUARIES

0

2

4

6

8

10

12

14

z / m

11 13 15

S / psu

-2 -1

v / m s-1

0.0 0.2 0.4

Ri

10-5 10-4 10-3

ε / W kg-1

0 0.4 0.8

lth / m

-2 0 2

ζth / m

10-4 10-2

N2 / s-2

N2

u*3 (κz)-1

v(a) (b) (c) (d)

ε

S

Ri

Figure 25.2: Microstructure profile taken during spring tides with anomalouslystrong stratification showing vigorous mixing throughout the water column,drop 6652, 23 Oct. 1995 19:01 UT. (a) Streamwise velocity from the ADCP andsalinity from the microstructure conductivity sensor (thick line, highly variable)as well as from the CTD, the latter Thorpe-sorted (thin line). (b) Squaredbuoyancy frequency from the regular CTD (thick) and from the microstructureT/C sensor (thin) as well as Richardson number from ADCP shear and CTD–N2 (shaded). (c) Measured dissipation rate and dissipation rate expected fromlaw-of-the-wall scaling. (d) Turbulent displacement (line) and Thorpe scale(shaded) based on the microstructure conductivity.

The dissipation method cannot possibly be valid at “large” Ri À 1/4, becauseinternal waves are expected to dominate. [Peters et al., 1988, Gregg, 1987] dis-cuss the topic in more depth. Lack of knowledge of Rf in (25.1) is a minorproblem because Rf is small compared to 1 [Osborn, 1980].

[Osborn, 1980] introduced (25.2) arguing that an upper bound for Kρ is

given by Rf (1 +Rf )−1

∼< 0.2. In practice it has been common to use

Rf (1 +Rf )−1

= 0.2 which can be justified by comparing Kρ with an eddydiffusivity of heat estimated independently from the thermal dissipation rateχ [Osborn and Cox, 1972, Oakey, 1982, Moum, 1990]. To assume a constantflux Richardson number is not appropriate in estuarine flow, however, be-cause Ri tends to be small. [Peters and Bokhorst, 2001] developed a model forRf = Rf (Ri) used in the following with limRi→0Rf = 0, limRi→∞Rf = 0.19

and Rf (1 +Rf )−1

= 0.22 at Ri = 0.25.With (25.1) the momentum flux vector becomes

~Jm = −ρKm ∂~u/∂z , (25.3)

25.2. TIDAL, FORTNIGHTLY AND LONGER-TERM VARIABILITY 5

related to the turbulent stress as ~τ = − ~Jm. The buoyancy flux has already beengiven above. As temperature effects are negligible in the Hudson River, B isproportional to the vertical turbulent salt flux

JS = −10−3Kρ ∂S/∂z , (25.4)

where the factor 10−3 converts practical salinity S to concentration units suchthat JS attains a unit of kg m−2s−1.

25.2 Tidal, fortnightly and longer-term variabil-

ity

Vertical profiles of density, velocity and measures of turbulent mixing differsignificantly between ebb and flood throughout the fortnightly tidal cycle (Fig-ure 25.3). This is a result of the superposition of the barotropic tidal forc-ing and the baroclinic forcing resulting from the longitudinal density gradient.These oppose each other during flood and add to one another during ebb. Thelateral straining of the density gradient by the combined tidal and residualflow tend to cause enhanced stratification during ebb and reduced stratificationduring flood (the “strain-induced periodic stratification” mechanism (SIPS) of[Simpson et al., 1990]. The SIPS process would suggest stronger mixing duringflood than during ebb, and this is what the Hudson River shows during neaptides. However, upon the progression from neaps to springs turbulent mixingcontinually reduces the stratification such that Richardson numbers eventuallybecome small throughout the water column even during ebb. Shear instabilityand vigorous turbulence ensue, and the strongest mixing now occurs during ebbrather than flood.

The tidal and fortnightly variability of flow and mixing is illustrated in theform of time-depth contour diagrams in Figure 25.31 and by mid-depth tidaltime series in Figures 25.4 and 25.5. During neap tides, Ri is low and ε is largeonly in the weakly stratified bottom layer during flood, while Ri is large andε is comparatively small in the halocline and also close to the bottom duringebb. During neap tides significant JS occurs only during flood. As a conse-quence of turbulent mixing, the stratification is reduced strongly toward springtides, and the shear is also reduced in such a manner that the flow through-out almost the entire water column becomes critical (Ri < 1/4), a feature firstobserved by [Geyer and Smith, 1987]. Figure 25.3h shows areas of sustainedsmall Ri < 1/4 and even Ri < 0.1 accompanied by large JS (and correspond-ingly large ε) throughout the water column. In the spring ebb regime densityand velocity profiles tend to be linear in depth with minor variations in Ri

[Geyer and Smith, 1987]. Profiles shown in Figure 25.2 conform to this descrip-tion when finescale fluctuations with vertical scales of the order of 1 m aredisregarded.

1See also Figure 5 of [Peters and Bokhorst, 2000] and Figures 5–7 of [Peters, 1997].

6 CHAPTER 25. PARTIALLY MIXED ESTUARIES

5 00.10.250.512∞5 00.10.250.512∞

0

4

8

12

16

z / m

-1.2 -0.6 0.0 0.6 1.2 -1.2 -0.6 0.0 0.6 1.2

0

4

8

12

16

z / m

10 14 18 22 26 10 14 18 22 26

0

4

8

12

16

z / m

0 90 180 270 360ΦSD / deg

10-6 10-5 10-4 10-3

0

4

8

12

16

z / m

0 90 180 270 360ΦSD / deg

10-6 10-5 10-4 10-3

v / m s-1v / m s-1

0

4

8

12

16

z / m

JS / kg m-2 s-1 JS / kg m-2 s-1

(e) Apr. 13−14, 1995(a) Apr. 8−10, 1995 − HUD3

S / psu

(b) (f)

S / psu

Ri

(g)(c)

Ri

(d) (h)

Figure 25.3: Tidal and fortnightly evolution of flow, stratification and mixingin the Hudson River. Contours as function of semidiurnal tidal phase ΦSD andheight above bottom z, April 8–10, 1995 (a–d), neap tide, and April 13–14,1995, (e–h), spring tide. Streamwise velocity (a, e), salinity (b, f), Richardsonnumber (c, g), and turbulent vertical salt flux (d, h).

25.2. TIDAL, FORTNIGHTLY AND LONGER-TERM VARIABILITY 7

(a) N2, springs

(b) N2, transition

(c) N2, neaps

(d) Vz2, springs

(e) Vz2, transition

(f) Vz2, neaps

(g) Ri, springs

(h) Ri, transition

(i) Ri, neaps

10 -4

10 -3

10 -2

10 -1

N2 ,

Vz2

/ s-2

10 -1

10 0

10 1

Ri

10 -4

10 -3

10 -2

10 -1

N2 ,

Vz2

/ s-2

10 -1

10 0

10 1

Ri

10 -4

10 -3

10 -2

10 -1

N2 ,

Vz2

/ s-2

0 90 180 270 360

ΦSD / deg

0 90 180 270 360

ΦSD / deg

10 -1

10 0

10 1

Ri

0 90 180 270 360

ΦSD / deg

Figure 25.4: Mid-depth squared buoyancy frequency, squared shear and gradientRichardson number versus semidiurnal tidal phase from 5 cruises.

The Hudson data have been averaged as function of normalized height abovebottom, z/H with local water depth H. They are displayed as a function ofnominal dimensional height above bottom, however, using an average depthof 16 m. Tidal variations of depth are thus suppressed in Figure 25.3. Figures25.3–25.5 use the semidiurnal tidal phase ΦSD introduced in [Peters, 1999]; ΦSD

accounts for both M2 and S2 tides. The ebb sector corresponds to 90◦ < ΦSD <270◦, while the flood sector is split into 0◦ ≤ ΦSD < 90◦ and 270◦ < ΦSD <360◦. Figures 25.4 and 25.5 display data from our five cruises of 1994/95 andthus allow an assessment of the longer-term variability.

Figure 25.4 shows that N2 and shear exhibit mostly semidiurnal variationswith minima during late flood and maxima during late ebb. Maximum V 2

z

migrate from late ebb during neaps to central ebb during springs. Ri exhibitsquarter-diurnal variations with flood and ebb minima throughout, spring tidesshowing much smaller Ri especially during ebb. The dissipation rate and thevertical salt flux change from semidiurnal variations with flood-tide maxima andebb-tide minima during neap tides to quarter-diurnal variations with ebb andflood maxima during spring tides (Figure 25.5).

The maximum mid-depth salt flux shown in Figure 25.5 is approxi-mately one gram, i.e. one teaspoon full, per square meter and second.[Peters and Bokhorst, 2001] demonstrate that, if such large JS acted in iso-

8 CHAPTER 25. PARTIALLY MIXED ESTUARIES

-----------++++++++++++-----------

------------++++++++++++------------

------------++++++++++++-------

-----

------------------------

----------------------

-----------

-----------

------------

------------

----------------------

------------------------

------------------------

--------------

------------

10 -7

10 -6

10 -5

10 -4

ε / W

kg

-1

10 -7

10 -6

10 -5

10 -4

ε / W

kg

-1

10 -7

10 -6

10 -5

10 -4

ε / W

kg

-1

0 90 180 270 360ΦSD / deg

(a) springs

(b) transition

(c) neaps0�

2�

4�

6�

10�4�

�× J� S

� / kg

m�-2

� s�-1

�

0�

2�

4�

6�

8�

10�

10�4�

�× J� S

� / kg

m�-2

� s�-1

�

0�

2�

4�

6�

10�4�

�× J� S

� / kg

m�-2

� s�-1

�

0� 90� 180� 270� 360ΦSD / deg�

(d) springs�

(e) transition�

(f) neaps �

Figure 25.5: Mid-depth viscous dissipation rate and turbulent salt flux versussemidiurnal tidal phase. Upper and lower confidence bounds for ε from all dataare plotted in the lower right hand corner.

lation, the spring-ebb turbulent salt flux would remove the stratification on thetime scale of a few hours, i.e. within a fraction of the tidal cycle. Spring ebbsare indeed special with respect to mixing in the Hudson estuary. They con-tribute roughly 1/3 of the total time-integrated fortnightly salt flux (

∫

fortJSdt)

even though significant spring ebb currents last less than 1/6 of a fortnight[Peters, 1999]. Flood tides throughout the fortnightly cycle carry most of theremainder of

∫

fortJSdt. Even though mixing in the halocline was rather weak

most of the time, the many small contributions added up significantly over theduration of a fortnight.

Figures 25.4 and 25.5 demonstrate a fair amount of repeatability in all vari-ables despite significant cruise-to-cruise variations. From cruise to cruise, N 2,V 2z and Ri varied more strongly during flood, especially spring floods, than dur-

ing ebb. The variability patterns of ε are more complex. The largest overall εoccurred during spring ebbs and showed a variability range of only a factor of3. Surprisingly, when the estimated turbulent salt flux data are integrated overfortnightly cycles, the results vary by only a factor of 2 [Peters, 1999]. Thisfinding has to be contrasted with an initial variability range of 0.5-m ε data

25.3. NOTES ON SOME OTHER ESTUARIES 9

spanning 4 decades.Two cruises in August and October 1995 were part of a comprehensive ex-

periment to probe the circulation of the Hudson estuary in all its aspects. Inte-gral balances of momentum and salt were worked out by [Geyer et al., 2000]on the basis mostly of moored observations. [Peters and Bokhorst, 2000,Peters and Bokhorst, 2001] show that the turbulent momentum and salt fluxesestimated from microstructure profiles were well compatible with the respectiveintegral balances and that the measured ε fit well within the overall energybalance.

25.3 Notes on some other estuaries

In the following we provide brief notes on similarities and differences in thevariability of flow and mixing between the Hudson River estuary and dy-namically similar sites. We focus on the Fraser River of British Columbia[Geyer and Smith, 1987, Geyer and Farmer, 1989] and Suisun Cutoff in north-ern San Francisco Bay [Stacey et al., 1999]. Both of these tidal estuaries displayconsiderable similarity with the Hudson. Common among all three sites are astrong ebb–flood asymmetry with enhanced shear on ebb and a jet on flood thatmigrates upward with time. Common also is the occurrence of the strongestturbulent mixing on ebbs in a regime with vertically near-constant shear andstratification.

No true turbulence measurements were done in the Fraser River in the late70s and early 80s. These observations are unique, however, in still being theonly estuarine water column measurements resolving the finescale flow withunsurpassed vertical and temporal resolution of 1 m and 1 s, respectively, invelocity and density. “Finescale” herein refers to variations on scales largerthan turbulent and smaller than tidal. These finescale measurements form thebasis for in-depth discussions on the role of internal waves in estuarine flow in[Geyer and Smith, 1987]. This topic has not been picked up since.

Compared with the lower Hudson, the lower Fraser River has similar channelwidth, slightly less depth, much larger average river flow, a shorter salt intru-sion, and comparable tidal velocities. At the beginning of ebb, a salt wedgeextends into the estuary with structure reminiscent of the neap tide in the Hud-son. Upon the intensification of the ebb tide, the Fraser displays a unique regimewith mixing even stronger than during spring tide in the Hudson as judged byits end result: the collapse of the salt wedge. [Geyer and Smith, 1987] demon-strate how the enhanced ebb tide shear causes strong shear instability in thepycnocline, a thickening of the pycnocline as a consequence of the turbulentmixing, and the eventual breakdown of the pycnocline. Interestingly, the bot-tom boundary layer is irrelevant in this event except that the bottom frictionallows the existence of strong vertical shear in the water column. After thebreakdown of the pycnocline salty water remains in a thin layer above the bot-tom. Neither the Hudson nor Suisun Cutoff display such remnant saline layers,and it appears that turbulence in the bottom boundary layer retained a more

10 CHAPTER 25. PARTIALLY MIXED ESTUARIES

important overall role in both of these sites during ebb as compared with theFraser. The Hudson remained diffusely stratified even during spring ebbs, whileSuisun Cutoff became completely unstratified at the end of a strong ebb seenby [Stacey et al., 1999].

While the barotropic and baroclinic forcing of the Fraser River is consider-ably stronger than in the Hudson, the tidal forcing is weaker in Suisun Cutoffthan in the Hudson with weaker initial stratification before its breakdown dur-ing a strong ebb. Yet, turbulence characteristics such as ε (or P ) and lengthscales have comparable magnitude and similar variability patterns.

More recent measurements in Liverpool Bay show an altogether differentscenario [Rippeth et al., 2001]. During one short time series taken in the tidallyforced fresh water-influenced regime, the strongest mixing occurred during floodwhen the stratification broke down completely. The breakdown of the stratifi-cation appears to be the result of vertical convection resulting from differentialadvection of heavier (saltier) over lighter (fresher) water during flood as partof the SIPS mechanism [Simpson et al., 1990]. In comparison, negative N 2 alsooccurs regularly during flood in the Hudson River, but it affects only the lowest2–3 m of the water column. The SIPS mechanism thus plays a much more dra-matic role in Liverpool Bay than in the Hudson or Suisun Cutoff. We concludethat the balance in the interplay of tidal and freshwater forcing and turbulentmixing in estuaries and dynamically similar regimes is quite delicate.

25.4 Turbulence closure simulation

Our closure modeling differs from similar efforts reported in this volume andelsewhere in that we make no effort to model the mean flow variables, focusentirely on turbulence variables, and attempt a fully quantitative 1:1 comparisonof observed and modeled dissipation rates and turbulent length scales, l. Theinclusion of l in the measurement–model comparison inclusion of proved to becritical toward optimizing model parameters. Direct quantitative comparisonsof simulated and observed turbulence variables can be meaningful only when theobserved and simulated mean flow variables very closely match each other. Thisis due to the extreme sensitivity of turbulence closure schemes to even minorvariations of the gradient Richardson number in the vicinity of the “critical”value near 1/4.

Achieving a close match of observed and modeled mean velocity and den-sity proved to be difficult in the estuarine flow of the Hudson River in whichnot only the baroclinic forcing in the momentum balance and the horizontaladvection of the density field are of first order importance, but which alsoshows large vertical advection owing to hydraulic processes at a constriction[Chant and Wilson, 2000]. Observations of the streamwise density gradient werequite limited, and vertical velocities are impossible to estimate.

Two-equation turbulence closures provide predictive differential equationsfor two turbulence variables in which the turbulence can evolve on its own timescale. We thus insist on forward-integrating the closure scheme. This is imple-

25.4. TURBULENCE CLOSURE SIMULATION 11

mented by averaging the observations as function of semidiurnal tidal phase andconsidering the tidal variations to be periodic, and further by integrating themodel outlined below over 3 tidal cycles, taking results from the last cycle. Thisapproach accounts for the time needed for the turbulence closure to “spin up”on its own, this time scale being of the order of 1–2 h. We keep the simulateddensity and velocity close to their observed counterparts by heavily “nudging”them toward the observations. With nudging time constants on the order of 15min other aspects of the mean field equations become unimportant and will notbe discussed herein.

We are using a variant of the k − ε closure modelof [Burchard and Baumert, 1995, Burchard and Baumert, 1998,Regener et al., 1997]. The two predictive differential equations for theturbulent kinetic energy, k, and for its dissipation rate are:

dk

dt−

∂

∂z

(

Km

σk

∂k

∂z

)

=(

Km V 2z − KρN

2)

− ε , (25.5)

∂ε

∂t−

∂

∂z

(

Km

σε

∂ε

∂z

)

= cε1(

KmV2z − cε3KρN

2) ε

k− cε2

ε2

k, (25.6)

where the eddy viscosity and eddy diffusivity follow from

Km = cµk2

ε,Kρ =

Km

σh. (25.7)

The model parameters listed on the lefthand side of Table 25.1 follow[Baumert and Peters, 2000]. Boundary conditions for k and ε are discussed in[Baumert and Radach, 1992] The parameters cε3 and R∞

f were considered ad-justable and were optimized by trial and error. The functional form of turbulentPrandtl number σh = Ri/Rf = σh(Ri) is that of [Schumann and Gerz, 1995]:

σh = σ0h exp

(

−Ri

σ0hR

∞f

)

+Ri

R∞f

. (25.8)

[Baumert and Peters, 2000] discuss the importance of steady state behav-ior for unbounded, spatially homogeneous, stratified shear flow (N 2=const,V 2z =const). Our model exhibits full steady state at realistic values of the

gradient and flux Richardson numbers, at Ri = Rsi and Rf = Rs

f , respec-tively (Table 25.1). Our function Rf = Rf (Ri) matches the laboratory dataof [Rohr et al., 1988] well at small Ri, lies below these at Ri ∼> 0.2, where itmatches the oceanic data of [Peters et al., 1988]. The turbulence simulatedthrough (25.5) and (25.6) is most sensitive to the value of Rs

i and less sensitiveto such variations of cε3 and R∞

f that are compatible with constant Rsi .

With near-optimal parameters, our model shows considerable skill in simu-lating the time-depth variability patterns of the turbulence as well as the mag-nitudes of ε and l. Figure 25.6 depicts modeled and simulated turbulence forthe observations of October 19, 1995, during neap tide. As the turbulent kineticenergy was not measured, we compute a turbulent length scale from the model

12 CHAPTER 25. PARTIALLY MIXED ESTUARIES

Table 25.1: Standard (left) and optimized (right) model parameters.Parameter Value Parameter Valuecµ 0.09 cε3 -1.25σk 1 R∞

f 0.1375

σε 1.11cε 1.5cε2 2 Rs

i 0.175σ0h 0.63 Rs

f 0.129

data by inverting the Taylor scaling [Taylor, 1935], ε = cε4 k3/2 l−1. Taylor

scaling is used explicitly in k − l turbulence closures and implicitly in k − ε

models. [Baumert and Peters, 2000] show that cε4 = c3/4µ . The simulated l can

be compared with the observed lth assuming l = 0.46 lth on the basis of labora-tory observations by [Rohr et al., 1987], results being shown in Figure 25.6e,f.Because of the way in which the observed Thorpe scale lth is computed, it isnot limited by z, and l and lth are not comparable in the lowest 2–3 m abovethe bottom.

0

4

8

12

16

z / m

0

4

8

12

16

z / m

0 90 180 270 360ΦSD / deg

0 90 180 270 360ΦSD / deg

0 90 180 270 360ΦSD / deg

(a) v (observed) / (m/s) (c) log epsilon (observed) / (W/kg)

v

S

(b) S (observed) / psu

10 14 18 22 26

(e) log turbulent length scale (observed) / m

−2 −1 0 1−9 −7 −5 −3210−1−2

(d) log epsilon (model) / (W/kg) (f) log turbulent length scale (model) / m

Figure 25.6: Observed and simulated turbulent mixing in the Hudson River onOct. 19, 1995, neap tide. Contours as function of semidiurnal tidal phase ΦSD

and height above bottom z. (a) Observed streamwise velocity (extrapolated tosurface and bottom), (b) observed salinity (extrapolated to surface and bottom),(c) observed and (d) simulated viscous dissipation rate, (e) observed (lth) and (f)simulated turbulent length scale (0.46 l). Wherever simulated variables attainminimum values allowable in the model, they are not contoured (white areas).

While Figure 25.6 depicts data from neap tide, the scatter diagrams of model

25.4. TURBULENCE CLOSURE SIMULATION 13

versus measurements shown in Figure 25.7 also includes data from spring tide.The displayed 1:1 model–measurement comparison includes ε and k. The “ob-

served” TKE (kobs) was calculated as (ε 2.8 lth)2/3

on the basis of Taylor scalingas outlined above. The limitation of the length scale by the distance to thebottom is taken into account. Figure 25.7 further depicts model and observedRichardson numbers in order to allow an assessment of the match betweenmodeled and observed mean flow variables. The figure displays correlation co-efficients (R) for all variables shown.

10-7

10-5

10-3

k mod

/ m

2 s-2

10-7 10-5 10-3 kobs / m

2s-2

(a) ...............

.

..

10-8

10-6

10-4

ε mod

/ m

3 s-2

10-8 10-6 10-4 εobs / m

3s-2

(b)

..................

0.0

0.8

1.6

arc

tan(

Ri m

od)

0.0 0.8 1.6arc tan(Riobs)

(c)

............

.

.

.. ..

..................

................

..

.....

.........

.

.

..

..................

..................

... ..

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Figure 25.7: Simulated versus observed (a) turbulent kinetic energy and (b)dissipation rate, and, (c), arc tangent of the gradient Richardson number inthe model versus that of the observed Ri; Oct. 19 and 23, 1995. Correlationcoefficient are indicated in the panels.

Noting that small Ri roughly corresponds to large ε and k we summarizethe quantitative model-observation comparison as follows.

• Except for large Ri, modeled ε visually match observed dissipation rateswell. When the data are restricted to Ri < 0.5 the mean ratio of εmod/εobsis 0.78 with a standard deviation of 0.73.

• Modeled and observed TKE at low Ri is also match each other well visuallyand numerically with mean kmod/kobs of 1.0 and standard deviation of 1.0.

• The typical model error in ε and k thus is a factor of 2 with mean error ofless then 25%. The model explains 50% of the observed variance in log εand log k.

• At large Ri, the model severely underestimates k and ε.

• The nudging does not keep model values of Ri as close to the observationsas one would like.

For conditions of small Ri and correspondingly intense mixing, and whendriven by N2 and V 2

z close to the observations, our closure produces realistic εand k. Whether the closure is realistic enough ultimately has to be tested inthe context of a full three-dimensional circulation model. Modeling estuarinecirculations places tough demands on the employed turbulence closure because

14 CHAPTER 25. PARTIALLY MIXED ESTUARIES

lateral advection, vertical mixing and stratification mutually affect each otherin positive and negative feedback loops. The failure of our closure at large Ri

is of minor consequence because the associated mixing is weak. We attributethis failure to internal waves resolved neither in the model nor in the observedmean shear. Internal gravity waves are ubiquitous in stratified fluids in generaland demonstrably omnipresent in the Hudson. In contrast to our judgment,[Stacey et al., 1999] associate the failure of turbulence closure at large Ri witha poor parameterization of turbulence transport.

Our modeled turbulence shows properties interesting with respect to flowphysics.

• In our model results we find an approximate production–buoyancy flux–dissipation balance at all depths most of the time except for transients thatbegin at the bottom at the onset of ebb and flood and migrate upwardwith time. These transients take the form of a sharp turbulence front.

• The magnitudes of the tendency and turbulence transport terms in theturbulent kinetic balance, on the lefthand side of (25.5), remain below 5%of the production term except during the transients.

• The tendency and transport terms are roughly of the same magnitude.

Our finding of the tendency term and the turbulence transport termin the TKE equation having similar magnitude calls modeling results of[Stacey et al., 1999] into question. In a model-observation comparison for Su-isun Cutoff in San Francisco Bay, they approached the problem of matchingmodeled to observed mean flow variables by inserting the observed velocity anddensity at each observation time into the model, keeping them constant, andletting the tendency term in the TKE equation converge to zero while retainingthe transport term. We replicated this procedure with an early version of ourown closure and found little similarity between model output and observations.

Finding a balance between net production and dissipation lends supportto our methods of estimating turbulent momentum and mass fluxes from mi-crostructure observations, (25.3) and (25.4) above. Finding this balance hasmore fundamental relevance, however. According to our model, the HudsonRiver sustains a production-dissipation balance even at very low Ri ∼< 0.1.This differs qualitatively from the laboratory stratified shear flow experimentsof [Rohr et al., 1987] in which the turbulence grows exponentially at Ri < 1/4.The observed ε confirms the lack of systematic growth of the turbulence even atthe lowest Ri [Peters and Bokhorst, 2001]. Hence, the Hudson River does notbehave like a homogeneous shear layer, and it does not display the “structuralequilibrium” of [Baumert and Peters, 2000].

Acknowledgments. The 1994/95 Hudson experiments were funded by theNational Science Foundation of the USA. H.P. gratefully acknowledges travelsupport by the European Commission for participation in the Cartum project.We enjoyed enlightening discussions with Cartum participants and with TamayOzgokmen.

25.4. TURBULENCE CLOSURE SIMULATION 15

Symbols Used in Chapter 25.

Symbol Meaning Units FirstAppearance

S practical salinity (psu) P. 2~u=(u, v) horizontal current vector m s−1 P. 2

u spanwise velocity component m s−1 P. 2v streamwise velocity component m s−1 P. 2T temperature oC P. 2C electrical conductivity S m−1 P. 2ε viscous dissipation rate W kg−1 P. 2g gravity m s−2 P. 2ρ density kg m−3 P. 2ρpot potential density kg m−3 P. 2N buoyancy frequency rad s−1 P. 2Vz vertical shear of ~u rad s−1 P. 2Ri gradient Richardson number P. 2lth Thorpe scale m P. 2P production of TKE W kg−1 P. 3B buoyancy flux W kg−1 P. 3Km eddy viscosity m2 s−1 P. 3Kρ eddy diffusivity of mass m2 s−1 P. 3Rf flux Richardson number P. 3~Jm vertical turbulent momentum flux Pa P. 5JS vertical turbulent salt flux kg m−2 s−1 P. 5ΦSD semidiurnal tidal phase deg P. 7k turbulent kinetic energy (TKE) J kg−1 P. 11σh turbulent Prandtl number P. 11

See Table 12 for further symbols.

16 CHAPTER 25. PARTIALLY MIXED ESTUARIES

Bibliography

[Baumert and Peters, 2000] Baumert, H. and H. Peters, Second moment closuresand length scales for stratified turbulent shear flows,J. Geophys. Res., 105, 6453–6468, 2000.

[Baumert and Radach, 1992] Baumert, H. and G. Radach, Hysteresis of turbulentkinetic energy in nonrotational tidal flows: A modelstudy, J. Geophys. Res., 97, 3669–3677, 1992.

[Burchard and Baumert, 1995] Burchard, H. and H. Baumert, On the performanceof a mixed-layer model based on the k− ε turbulenceclosure, J. Geophys. Res., 100, 8523–8540, 1995.

[Burchard and Baumert, 1998] Burchard, H. and H. Baumert, The formation of estu-arine turbudity maxima due to density effects on thesalt wedge. A hydrodynamic process study, J. Phys.

Oceanogr., 28, 309–321, 1998.

[Busch, 1977] Busch, N. E., Fluxes in the surface boundary layerover the sea, in Modelling and Prediction of the Upper

Layers of the Ocean, , edited by E. B. Kraus, pp. 72–91. Pergamon, New York, 1977.

[Cameron and Pritchard, 1963] Cameron, W. M. and D. W. Pritchard, Estuaries, inThe Sea: Ideas and Observations on Progress in the

Study of the Seas, , edited by M. N. Hill, volume2: The Composition of Sea-Water, Comparative andDescriptive Oceanography, pp. 306–324, New York,1963. Wiley-Interscience.

[Chant and Wilson, 2000] Chant, R. J. and R. E. Wilson, Internal hydraulicsand mixing in a highly stratified estuary, J. Geophys.

Res., 105, 14,215–14,222, 2000.

[Geyer and Farmer, 1989] Geyer, W. R. and D. M. Farmer, Tide-induced varia-tion of the dynamics of a salt wedge estuary, J. Phys.

Oceanogr., 19, 1060–1072, 1989.

[Geyer and Smith, 1987] Geyer, W. R. and J. D. Smith, Shear instability ina highly stratified estuary, J. Phys. Oceanogr., 17,1668–1679, 1987.

[Geyer et al., 2000] Geyer, W. R., J. H. Trowbridge, and M. M. Bowen,The dynamics of a partially mixed estuary, J. Phys.

Oceanogr., 30, 2035–2048, 2000.

17

18 BIBLIOGRAPHY

[Gregg, 1987] Gregg, M. C., Diapycnal mixing in the thermocline:A review, J. Geophys. Res., 92, 5249–5286, 1987.

[Gregg et al., 1985] Gregg, M. C., H. Peters, J. C. Wesson, N. S. Oakey,and T. J. Shay, Intensive measurements of turbu-lence and shear in the equatorial undercurrent, Na-

ture, 318, 140–144, 1985.

[Hansen and Rattray, Jr., 1965] Hansen, D. V. and M. Rattray, Jr., Gravitational cir-culation in straits and estuaries, J. Mar. Res., 23,104–122, 1965.

[Moum, 1990] Moum, J. N., The quest for kρ — prelimiary resultsfrom direct measurements of turbulent fluxes in theocean, J. Phys. Oceanogr., 20, 1980–1984, 1990.

[Oakey, 1982] Oakey, N. S., Determination of the rate of dissipationof turbulent energy from simultaneous temperatureand velocity shear microstructure measurements, J.

Phys. Oceanogr., 12, 256–271, 1982.

[Osborn, 1980] Osborn, T. R., Estimates of the local rate of verti-cal diffusion from dissipation measurements, J. Phys.

Oceanogr., 10, 83–89, 1980.

[Osborn and Cox, 1972] Osborn, T. R. and C. S. Cox, Oceanic fine structure,Geophys. Fluid Dyn., 3, 321–345, 1972.

[Peters, 1997] Peters, H., Observations of stratified turbulent mix-ing in an estuary. Neap-to-spring variations duringhigh river flow, Estuar. Coast. Shelf Sci., 45, 69–88,1997.

[Peters, 1999] Peters, H., Spatial and temporal variability of turbu-lent mixing in an estuary, J. Marine Res., 57, 805–845, 1999.

[Peters and Bokhorst, 2000] Peters, H. and R. Bokhorst, Microstructure obser-vations of turbulent mixing in a partially mixed es-tuary. I: Dissipation rates, J. Phys. Oceanogr., 30,1232–1244, 2000.

[Peters and Bokhorst, 2001] Peters, H. and R. Bokhorst, Microstructure observa-tions of turbulent mixing in a partially mixed estu-ary. II: Salt flux and stress, J. Phys. Oceanogr., 31,1105–1119, 2001.

[Peters et al., 1988] Peters, H., M. C. Gregg, and J. M. Toole, On the pa-rameterization of equatorial turbulence, J. Geophys.

Res., 93, 1199–1218, 1988.

[Pritchard, 1952] Pritchard, D. W., Estuarine hydrography, Adv. Geo-

phys., 1, 243–280, 1952.

[Pritchard, 1954] Pritchard, D. W., A study of the salt balance ina coastal plain estuary, J. Mar. Res., 13, 133–144,1954.

[Pritchard, 1956] Pritchard, D. W., The dynamic structure of a coastalplain estuary, J. Mar. Res., 15, 33–42, 1956.

BIBLIOGRAPHY 19

[Pritchard, 1967] Pritchard, D. W., Observations of circulation incoastal plain estuaries, in Estuaries, , edited by G. H.Lauf, pp. 37–44. Amer. Assoc. Advanc. Sci., 1967.

[Rattray and Hansen, 1962] Rattray, M. and D. V. Hansen, A similarity solutionfor the circulation in an estuary, J. Mar. Res., 20,121–133, 1962.

[Regener et al., 1997] Regener, M., B. Szilagyi, and H. Baumert, Thegeneric 1-D turbulence-SPM model, Tech. rep. 48, 46pp., Proudman Oceanographic Laboratory, Bidston,UK, 1997.

[Rippeth et al., 2001] Rippeth, T. P., N. R. Fisher, and J. H. Simpson, Thecycle of turbulent dissipation in the presence of tidalstraining, J. Phys. Oceanogr., 31, 2458–2471, 2001.

[Rohr et al., 1987] Rohr, J. J., K. N. Helland, E. C. Itsweire, and C. W.Van Atta, Turbulence in a stratified shear flow: Aprogress report, in Turbulent Shear Flows, , editedby F. Durst, New York, 1987. Springer.

[Rohr et al., 1988] Rohr, J. J., E. C. Itsweire, K. N. Helland, and C. W.Van Atta, Growth and decay of turbulence in a sta-bly stratified shear flow, J. Fluid Mech., 195, 77–111,1988.

[Schumann and Gerz, 1995] Schumann, U. and T. Gerz, Turbulent mixing in sta-bly stratified shear flows, J. Appl. Meteorol., 34, 33–48, 1995.

[Simpson et al., 1990] Simpson, J. H., J. Brown, J. Matthews, and G. Allen,Tidal straining, density currents, and stirring in thecontrol of estuarine stratification, Estuaries, 13, 125–132, 1990.

[Stacey et al., 1999] Stacey, M. T., S. G. Monismith, and J. R. Burau,Observations of turbulence in a partially stratifiedestuary, J. Phys. Oceanogr., 29, 1950–1970, 1999.

[Taylor, 1935] Taylor, G. I., Statistical theory of turbulence. PartsI–IV, Proc. R. Soc. London A, 151, 421–478, 1935.

[Thorpe, 1977] Thorpe, S. A., Turbulence and mixing in a Scottishloch, Philos. Trans. R. Soc. London A, 286, 125–181,1977.

[Unesco, 1981] Unesco, editor, The Practical Salinity Scale 1978 and

the International Equation of State of Seawater 1980,volume 36 of Unesco Technical Papers in Marine Sci-

ence. Unesco, 1981.