JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. C5, PAGES 10,543-10,554, MAY 15, 1998

Comparing the performance of the Mellor-Yamada and the two-equation turbulence models

Hans Burchard

Joint Research Centre, Space Applications Institute, Ispra, Italy

Ole Petersen International Research Centre for Computational Hydrodynamics, Danish Hydraulic Institute H0rsholm, Denmark

Tom P. Rippeth School of Ocean Sciences, University of Wales, Bangor, Gwynedd

Abstract. The aim of this paper is to systematically compare k-e and Melior- Yamada two-equation turbulence models. Both models include prognostic equations for turbulent kinetic energy and a length scale related parameter which are used to calculate eddy viscosities and vertical diffusivities. The results from laboratory experiments, using mixed and stratified flows, are simulated in order to systematically compare and calibrate the models. It is shown that the Monin- Obukhov similarity theory is well represented in both models. The models are used to simulate stratified tidal flow in the Irish Sea, and the results show that the k-e models generally predict a larger phase lag between currents and turbulent dissipation, in the bottom boundary layer, than the Mellor-Yamada models. The comparison between the model results and field measurements, of the rate of dissipation of turbulent kinetic energy, shows that both models require modification through the inclusion of an internal wave parameterization in order that they are able to correctly predict the observed levels of turbulent dissipation. As the main result, it is shown that the choice of the stability functions, which are used as proportionality factors for calculating the eddy viscosity and diffusivity, has a stronger influence on the performance of the turbulence model than does the choice of length scale related equation.

1. Introduction

The vertical structure and thereby the vertical fluxes of organic matter and primary production within estu- aries and coastal seas are largely controlled by the ver- tical mixing and therefore the vertical distribution of turbulence in the water column [Denman and Gargett, 1995]. First-order accounts of the evolution of the den- sity structure have been given in terms of bulk models which are closed by simplifying assumptions as, for ex- ample, constant mixing efficiencies [Kraus and Turner, 1967] or prescribed velocity profiles [Simpson et al., 1991]. Such models, however, do not resolve the ver- tical structure and cannot therefore be used to inves-

tigate the details of mixing and vertical fluxes in the water column.

Copyright 1998 by the American Geophysical Union.

Paper number 98JC00261. 0148-0227/98/98JC-00261509.00

In the last 20 years, differential models which resolve the vertical structure of turbulence have been estab-

lished in ocean and atmospheric sciences. Among them, so-called two-equation models, using two equations for calculating turbulent quantities, have been proven to be a good compromise between accuracy and efficiency. The two most well known amongst these are the k-e and the Mellor-Yamada model (from now on simply abbre- viated by "MY model"), are compared in this paper and validated using laboratory and field data.

It should be noted that usually different versions of the Mellor-Yamada model are referred to by indicating the level of the model. As this notation is not always unique, we simply refer to the number of differential equations (zero, one, or two) used for a model. The two- equation model under investigation here is the current standard which is often referred to as the level 2.5 model

[see Galperin et al., 1988; Mellor, 1989]. Several authors have carried out intercomparisons of

turbulence models. Martin, [1985] made a compari- son between two differential models (the algebraic zero-

10,543

10,544 BURCHARD ET AL.: TWO-EQUATION TURBULENCE MODELS

equation MY and the one-equation MY model) and bulk models. He found, by simulating the annual cycle of the mixed layer depth recorded at two weathership stations in the Pacific, that the differential models significantly underpredicted the mixed layer depth. By comparing the two-equation Mellor and Yamada [1982] model with a bulk model and a nonlocal model, Large and Crawford [1995] found that the nonlocal model could best predict mixing of the mixed layer deep into the thermocline. In order to address the deficiencies of differential mod-

els, other authors [Mellor, 1989; Kantha and Clayson, 1994] have added parameterizations of mixing in the thermocline, to simulate the effects of shear instability or internal wave breaking.

Burchard and Baumert [1995] have compared the standard and the advanced k-s models with algebraic turbulence parameterizations, among them the zero- equation MY model. The k-s models are reported to be easily tunable and give better agreement with observed temperature profiles from the northern North Sea, over a large tuning parameter range. In the work by Bur- chard and Baumert [1995], as well as in earlier pub- lications [e.g., Frey, 1991], the inherent numerical in- stability of zero-equation turbulence parameterizations has been highlighted. Dissipation measurements in the Irish Sea (which will also be simulated in this paper) have been used by Simpson et al. [1996] for intercompar- ing different versions of zero- and one-equation Mellor- Yamada models. They found that the measured dis- sipation rate in the stratified region could only be re- produced by including an internal wave parameteriza- tion. This was later confirmed by Luyten et al. [1996b] who used different internal wave parameterizations in order to enable the k-s model under investigation to predict the measured dissipation rates in the strongly stratified region of the flow. One of the internal wave parameterizations presented by Luyten et al. [1996b] will be adopted for the study presented here. O. Pe- tersen (Numerical models of wind induced mixing in stably stratified flows, submitted to Continental Shelf Research, 1997) made a comparison between a standard k-s model without stability functions and the quasi- equilibrium MY model, restricted to stably stratified shear driven flow, but included also a two-dimensional flow with horizontal gradients, and was able to obtain similar results with both models. In a theoretical paper, H. Baumert et al. (On k-1 and k-s models for shallow seas: Review and outlook, submitted to Journal of Ma- rine Systems, 1997) analysed zero-dimensional versions (neglecting the diffusion terms in the turbulence equa- tions) of two-equation k-s and MY models. They show the equivalence of these models by transforming them into each other with only slightly adjusted empirical parameters.

Mellor and Yamada [1982] have criticized the k-• ap- proach arguing that it is "fundamentally wrong" to use the Kolmogorov scale r/= y3/4/•1/4 (and thus •) for cal- culating the macro length scale L. In principle, r/can

only be directly related to L if the turbulence is in a spectral equilibrium. But as both models use only two parameters for the characterization of the turbulence spectrum and therefore must rely on this principle, Rodi [1987, p. 5311] states that this discussion is "rather aca- demic because both equations are fairly empirical and, with the constants suitably adjusted, perform in a sim- ilar manner". Moreover, both of the models are based on the same relation (11) between L and •. These sim- ilarities are often hidden in the way the equations are presented.

A systematic intercomparison, including field data, between two-equation k-• and MY models using the same stability functions suggested by Galperin et al. [1988] has, however, not yet been presented. It is our intention that this paper will fill the gap. One obstacle in directly comparing k-e and MY models is the differ- ent notation traditionally used for both of the models. We have therefore decided to translate the MY model

into k-e notation and add the corresponding MY no- tation for those who are more familiar with the latter models.

After presenting the two models in this common no- tation, boundary conditions (section 2), a discussion of stability functions (section 3) and discretization prin- ciples (section 4) are given. The performance of the models will then be compared for basic test cases. These are a purely wind-driven and a purely pressure gradient-driven barotropic channel flow (section 5.1), a wind-driven entrainment of a mixed layer into stably stratified flow (section 5.2) and a study of the Monin- Obukhov similarity (section 5.3). Finally, a realistic scenario of stratified tidal flow in the Irish Sea is sim-

ulated (section 6). In this case the model results are compared to in situ measurements of dissipation rate.

2. Basic One-Dimensional Mixed Layer Model

Assuming that all gradients with respect to x and y vanish, that is, that the flow is homogeneous in the horizontal, the rotational hydrodynamic equation can be written as

Otu - O• ((•'t 4- •,)O•u) - fv = F• (1)

O,v- + + =

where u and v are the velocity components with respect to x and y, vt is the eddy viscosity, v is the molecular viscosity, z is a Cartesian coordinate, positive in the up- ward direction, and f is the Coriolis acceleration. The right-hand side terms Fu and Fv denote the prescribed barotropic pressure gradient as a function of time.

The density distribution p is determined by the tem- perature T and the salinity $ which are modeled as follows

O•T- Oz ((• + •')OzT) = F• (3)

+ = rs, (4)

BURCHARD ET AL' TWO-EQUATION TURBULENCE MODELS 10,$45

where v• is the eddy diffusivity and p' and p" the molec- ular diffusivities and FT and F$ the sources and sinks of T and $, respectively.

If a linear equation of state is used,

p = p0 T0) $0),

where p0, To, and S0 are reference values and fit and •$ constant expansion coefficients, then (after neglect- ing sources, sinks, and molecular viscosities) an approx- imate equation for the buoyancy

can be derived

_ p - p0 (6) P0

Orb - (•Oz•) = 0. (7)

In modelling of turbulence, the turbulent kinetic en- ergy per unit mass k is an important quantity. It can be modeled as

Otk - Oz(VkOzk) -- P + B - e, (8)

where e is the dissipation rate of turbulent kinetic en- ergy. The turbulent kinetic energy k and the turbulence intensity q, which is commonly used in Mellor-Yamada models, are related by k = q2/2. For the calculation of the eddy diffusivity vk see section 3. The shear produc- tion P and the buoyancy production B are functions of the shear frequency M and the Brunt-Viiis/il/i frequency N, respectively,

= = (9) with

M •' = (Ozu) •' + (O•v) •' N •' = O•b. (10)

To calculate the eddy viscosity and diffusivity, a quantity related to the length scale of turbulence is re- quired. By dimensional arguments it is defined as

L= (cø,) s k•/•' (11) 0 is a constant (see section 3.3) The value where c, .

for the proportionality factor on the right-hand side of

(11) is determined to be (cø,) s in order that the log- arithmic law in the constant stress boundary layer is fulfilled. It can be derived from the relations P - e

2 (constant (turbulence in local equilibrium), •t0zu - u, stress), •t = nu,z' (linear eddy viscosity with z' as dis- tance from the bottom), L - nz' (linear length scale) and •t - c•,•1/2L (relation of Prandtl and Kolmogorov, see also (14)). This macro length scale describing the size of the largest turbulent eddies is equivalent to the master length scale I used by Mellor and Yamada [1974, 1982] and thus (11) is equivalent to (25c) of Melior and Yamada [1982].

The calculation of this length scale establishes the fundamental difference between the Mellor-Yamada and

the k-e model. In the latter model a transport equation for the dissipation rate e is calculated from which L is derived afterwards using (11),

- = + - where •e is the eddy diffusivity of e (see section 3). The length scale related equation introduced by Mellor and Yamada [1982] is presented here in a form which is mathematically identical to the original but allows a direct comparison with the k-e model. This equation calculates the product kL from which L can be easily derived afterwards

Ot(kL) - O• (•q:Oz(kL)) - L (c•:xP + c•:sB - c•:•.e) , (13)

where • is the eddy diffusivity of kL (see section 3). The right-hand sides of (12) and (13) appear to be nor- malized linear combinations of P, B, and e, including empirical coefficients specified in section 3.

Knowing k and L, the eddy viscosity and diffusivity can be calculated using the relationship of Kolmogorov and Prandtl

' =c•kl/•'L (14) •t = c•kl/2L •t , I

where c, and c, are stability functions (see section 3.3). These eddy viscosities and stability functions cor- respond to KM, KH, $M, and $H used in MY in the following way:

' •'M -- C1•/21/2 SH ' 21/2 KM = l,'t KH =1,' t -- Cl• / ß

In order to include the limiting effect of stable strat- ification, Galperin et al. [1988] found it necessary to introduce an upper limit for the macro length scale L in stably stratified flows

L •. < 0.56k N •. _ N• ' > 0. (15) Applied to k-e models, this corresponds to a lower limit for the dissipation rate e, which can be calculated using (11),

e •' _> 0.045k•'N 2 N •' > 0. (16) The constraints (15) and (16) describe the transition of large turbulent eddies into internal waves, in the pres- ence of stable stratification [see Luyten et al., 1996a]. Luyten et al. [1996b] also found that to ensure non- vanishing dissipation in stabily stratified regions it is necessary to set the lower limit of t,urbulent kinetic en- ergy. Here we use

k _• kmin - 7.6 x 10 -6 m •' s -•' (17)

for all our computations, a value which is tuned in order to give a realistic prediction for the stratified region in the Irish Sea simulation (see section 6). In stratified situations, this limitation of k can be interpreted as a crude parameterization of internal wave breaking which produces turbulence.

10,546 BURCHARD ET AL.' TWO-EQUATION TURBULENCE MODELS

Boundary conditions for L at the bottom and at the surface are

L = nzo (18)

with the von K•rm•n constant n - 0.4 and the bot-

tom or surface roughness length z0, respectively. The boundary conditions for k are derived from the assump- tion that a constant stress layer exists near the bound- aries. Deviating from the widely used equivalent Dirich- let condition (k is proportional to the square of the friction velocity), we use a more generally applicable no-flux condition

U•Ozk = O. (19)

At the bottom boundary and at a wind forced surface boundary we found in all our numerical experiments negligible differences between (19) and the Dirichlet condition. However, in pressure gradient-driven channel flow without wind forcing only condition (19) guaran- tees that the eddy viscosity converges to a small value near the surface (see section 5.1). This approach can be generalized to include wave breaking [see Craig and Banner, 1994] where the injection of turbulent kinetic energy through the sea surface is modeled by a nonho- mogeneous Neumann condition. Using equation (11), boundary values for either z or kL can be obtained from (18) and the resulting k at the boundary.

3. Specifications of the Turbulence Models

In this section the differences between the models are

specified and the empirical parameters are given. The stability functions should in principle be independent of the actual model, while their actual calibration is not, and this is discussed in section 3.3.

3.1. The k-z Model

The eddy diffusivities for k and z are calculated as

• - CU kX/2L (20)

c__• kX ,• - /2L (21)

with the Schmidt numbers a• and a,. Parameter a• is generally treated as unity (see Table 1), while a, de- pends on other constants'

m 1.08. (22) =

The values of the other constants are taken from Rodi

[1980] and are shown in Table 1. The value of c•s is dependent on the actual stability functions and is used as a calibration parameter (see section 5.2).

Table 1. Constants for the k-z Model

Constant Value

c•x 1.44 c•2 1.92 rr• 1.0 o 0.5562 cp

3.2. Mellor-Yamada Model

The eddy diffusivities for k and kL are calculated as

• - c•k•/2L (23)

YL - cLk•/2L (24) with the parameters c• - cœ. It is shown by Mellor and Yamada [1982] that these parameters are not indepen- dent

c• - cœ - (cøu) 3 ( l (E2 + l) ) •0.285. (25) 1,•2 • -- CL1 For those who are more acquainted with the Mellor-

Yamada model it would be helpful to note that the pa- rameters B•, E•, $q, and S• can be written as B1 - 2•'5(cø•)-• - 16.44, E• - 2cr• - 2crs- 1.8, and $q - S• - c•/2 •/• - c•/2 •/• - 0.2.

The parameter cœ2 is, in MY models, a function of the macro length scale L and a prescribed barotropic length scale L z

cœ2 - 1 + E2 ß (26)

The formulation of the prescribed length scale Lz will be discussed in section 5.1. The introduction of this

integral quantity Lz is necessary in MY models in order to ensure the logarithmic velocity distribution in steady state boundary layers [see Mellor and Yamada, 1982]. The remaining constants which are taken from Mellor and Yamada [1982] are given in Table 2.

3.3. Stability Functions

The role of the stability functions is to correct the eddy viscosity and diffusivity for further effects of strati- fication (stratification effects are already included in the buoyancy production term B on the right-hand sides of the k, the •, and the kL equations). Stability functions generally damp turbulent exchange for stable stratifica- tion and enhance turbulent exchange for unstable strat- ification. Mellor and Yamada [1974] introduced a set of stability functions depending on two non-dimensional parameters for stratification and shear

L2N2 (27) O•N -- •-

BURCHARD ET AL.- TWO-EQUATION TURBULENCE MODELS 10,547

Table 2. Constants for the Mellor-Yamada Model

Constant Value

cLx 0.9 cLs 0.9 E2 1.33 0 0.5562

L2M2 (28) O•M -- -•- ß The corresponding parameters in the MY literature are related to these as GM -- 0.5aM and GH -- --0.5aN. This set of stability functions has been shown to be mathematically unstable [see Deleersnijder and Luyten, 1994].

A set of quasi-equilibrium stability functions has been introduced by Galperin et al. [1988]. These functions were derived from the Mellor and Yamada [1974] sta- bility functions by assuming a local turbulence equilib- rium, that is, P + B - e only in the stability functions.

!

By doing so, stability functions c• and % are obtained which only depend on aN are mathematically stable.

With the constraints -0.0466 _< aN _< 0.56, the quasi-equilibrium stability functions read (see Figure 1)

cø• + 2.1824N c• = 1 + 2•-.4-•'••.124•v (29)

0.6985 , (30)

+ 17.34c•N

0 is the momentum stability function for neutral where c, flow (see Table 1). This set of stability functions is similar to those proposed by Luyten et al. [19964] to be used with k-e models.

Simple constant stability functions have been used in connection with k-e models for a long time [see Rodi, 1980]. Here we adopt that principle for some of our k-e

' from (29) and model calculations by using c, and c, (30) with aN -- 0, and thus neglect the influence of stratification on the stability functions.

4. Discretization

For the discretization the depth is divided into a num- ber of equidistant intervals. The discrete values for u, v, T, and S are located at the centers of the intervals, and the turbulent quantities k, L, e, vt, v•, N, P, B, c,, and

I

c, are positioned at the interfaces of the intervals. The staggering of the grid allows for a second-order approx- imation of the vertical fluxes of momentum and tracers

without averaging. However, for the vertical fluxes of k, L, and e, averaging of the eddy diffusivities is nec- essary. This is only problematic for the fluxes near the surface and the bottom, where viscosities at the bound- aries have to be considered for the averaging. These

can be derived from the boundary values for k and L 0 However, for e by applying (14) in connection with c•.

near the bottom it is necessary to use vt - u4,/e (where e has to be averaged) in order to achieve a good numer- ical approximation for a coarse vertical resolution [see $telling, 1995].

The turbulent quantities k, kL, and e are positive by definition. The conservation of positivity has been proven mathematically by Mohammadi and Pironneau [1994] for the barotropic k-e equations. In order to guarantee this in the discrete equations, the diffusion in these equations is treated implicitly, which leads to a tridiagonal linear system of equations. Moreover, the quasi-implicit approach of Patankar [1980] has been used for the sink terms. Taking the k equation as an example, the sink terms are multiplied with kn+l/k n, where n + I denotes a value on the "new" time level and n a value on the "old" time level. The sink terms

are then added to the main diagonal of the tridiagonal linear system and guarantee a positive solution for k. It should be noted that the buoyancy production B is a sink for stable stratification and a source for unstable

stratification.

5. Numerical Experiments

5.1. Barotropic Channel Flow

As basic comparative tests for the performance of the models, two steady state barotropic channel flow sit- uations, including comparisons with data, have been chosen: A surface stress forced flow which induces a

constant stress over the vertical (Couette flow) and a pressure gradient-driven flow (eg., tidal and river flow).

The main objective of studying these barotropic sit- uations is to choose a length scale Lz which has to be prescribed for the Mellor-Yamada model (see section 3.2). Near the surface and bottom, the law Lz - nz •, where z • is the distance from surface or bottom, should

4.0 _- ._ Galperin et al. [1988] stability functions for

i

- i

• Momentum _ i

- , Tracer i

_

.

_

_

3.5

3.0

2.5 2.0

1.5

1.0

0.5

0.0

-0.2 0.0 0.2 0.4 0.6

unstable (7. N stable Figure 1. Quasi-equilibrium stability functions ac- cording to Galperin et al. [1988].

10,$48 BURCHARD ET AL.: TWO-EQUATION TURBULENCE MODELS

be considered in any case. The classical way in doing this is to assume a parabolic profile of Lz [see Mel!or and Yamada, 1982]'

dsdb L•-•• (31)

ds

where ds is the distance from the surface and dv the distance from the bottom. Another function that fulfills

these requirements is

Lz - n min (dr, ds). (32)

The nondimensional results for the k-e and the two

versions of the MY model for k, e, and vt are shown in Figure 2.

It can be shown that for the Couette flow situation

the value of k/c, •/2 will be constant. Therefore the Ga!perin et al. [1988] stability functions result in a con- stant k distribution over the whole water column. This

is a consequence of the neglect of shear influence on the stability functions and is a simplification of the Cou- ette flow considered here. However, as the main task of

this paper is the intercomparison of models, we do not intend to change further the stability functions.

In both cases the MY model results for e and vt de- pend strongly on the choice for the length scale Lz. While the eddy viscosity calculated with version (32) is close to the data measured by Telbany and Reynolds [1982], the other version (31) underestimates the value significantly. Similarly for the pressure gradient-driven flow, data by Jobson and Sayre [1970] and Ueda et al. [1977] is significantly underestimated using (31). There- fore version (32) seems to be the more appropriate model for Lz, at least for well-mixed channel flow.

On the other hand, the k-e model overestimates the maximum eddy viscosity measured by Telbany and Reynolds [1982]. A close look at their data, however, re- veals that the flow in their experiment was probably not fully developed in the center. In the pressure gradient- driven experiment, the measured data are slightly un- derestimated by the k-e model. It should be noted for the pressure gradient-driven flow that the k-e model would result in an unrealistic, nearly constant eddy vis- cosity for the upper half of the water column, if the classical boundary condition for stress-driven boundary

l layers k -u, is applied instead of (19).

0 ß 0 ß Data from Telbany and Reynolds [1982]

-0.2 ß ß

-0.4 ß

-0.6 ß -0,8

-1.0 ......... i,,,,,,,,,i,,,,,,,,,, ......

ß Data from Telbany and Reynolds [1982] I

0.0

-0.2

-0.4

-0.6

-1.0

i

!

// i/

liiiiliiiiillillitiillllillrlliilllilll

0 10 20 30 40 0.00

œ / (u.D

0 1 2 3 4 0.05 0.10

k / u. 2 v t / (u.D)

Data from Nakagawa ©t al. [1975] Data from Jobson and Sayre [1970] Data from Ueda et al. [1977] Data from Nakagawa ©t al. [1975] ß

xxxxx x i

i !

. ß ß

0.15

ß

0 1 2 3 0 10 20 30 40 0.00 0.04 0.08

k / u. 2 e:/(u.3D -1) v t / (u,D)

Figure 2. Simulations of barotropic open channel flow with comparison against data. (top) Stress-driven Couette flow. Data from Telbany and Reynolds [1982] (turbulent kinetic energy and eddy viscosity). (bottom) Pressure gradient-driven flow. Data from Nakagawa et al. [1975] (turbulent kinetic energy and dissipation rate), Jobson and Sayre [1970], and Ueda et al. [1977] (both eddy viscosity). Thick line, k-e model; thin line, MY model, Lz = n min(ds,dv); dashed line, MY model, Lz -nds * d•/(ds + d•). Read 2.6E-4 as 2.6 x 10 -4.

BURCHARD ET AL.' TWO-EQUATION TURBULENCE MODELS 10,549

5.2. Wind-Induced Deepening of a Mixed Layer, Model Calibration

The supression of turbulence by stable density strat- ification is a major feature in upper ocean mixed layer dynamics. Turbulence models have to reflect this prop- erty of turbulence in a quantitatively correct way. The k-e models can be calibrated by tuning the parameter ce• [see Burchard and Baumert, 1995]. This is not the case for MY models, however, as numerical experiments show that the mixed layer depth predicted does not sig- nificantly depend on this parameter. The constant and the quasi-equilibrium k-e model will therefore be cali- brated to the MY model.

The experiments by Karo and Phillips [1969] will be used as the calibration scenario. In this experiment, a wind-induced mixed layer penetrates into a stably strat- ified fluid. Price [1979] suggested a solution for the evo- lution of the mixed layer depth Dm based on a constant Richardson number

Din(t)- 1.05u,N•-l/2tl/2, (33)

where No is the constant initial Brunt-V/•is/•l/• frequency. This curve is shown for ocean dimensions (u, - 10 -2 m s -1, No - 10 -2 s -x) in Figure 4 [see also Deleersnijder and Luyten, 1994]. In order to compare the models, the entrainment rate E - OtD,• averaged between t - 10 hours and t - 30 hours was calculated. By doing this, the effect of initial instabilities of the model after the

sudden onset of the surface stress are excluded from the calibration. In order to resolve the entrainment rate suf-

ficiently, a fine vertical resolution of Az - 0.25 m and a time step of At - 30 s was chosen. The calibration

2.6E-4

2.4E-4

'• 2.2E-4

2.0E-4

1.8E-4 -

1.6E-4

-2.0

(• Constant k-• - model • Quasi-equilibrium k-e - model

Quasi-equilibrium MY model

.... Price [1979], empirical

__ •ji._ _ • __ ___-H- -__.__-'-- ---'-

.

-1.5 -1,0 -0.5 0.0 0.5

Ce3

Figure 3. Simulation of wind-induced mixed layer deepening: Mean entrainment rate between t - 10 hours and t - 30 hours as a function of ce3 calcu- lated with the constant k-e and the quasi-equilibrium k-e model. The lines show the empirical value (dashed) and the value computed with the quasi-equilibrium MY model.

4O

35

3O

25

2O

C) constant k - • - model 0 quasi-equilibrium k - • - model ß quasi-equilibrium MY - model

Price [1979], empirical

Mixed layer depth

Entrainment rate

0 10 20 30 Time / h

1E-3

9E-4

8E-4

7E-4

6E-4

5E-4

4E-4

3E-4

2E-4

1E-4

OE+O

rn

Figure 4. Simulation of wind-induced mixed layer deepening: Development of mixed layer depth (MLD) and entrainment rate E as a function of time for differ- ent models. Read 1E-3 as I x 10 -3.

results in values which are ce3 • -0.4 for both the con- stant and the quasi-equilibrium k-e model. As shown in Figure 3, the quasi-equilibrium version is much less sen- sitive for c•3 k -0.7 than the constant version. Figure 4 shows that the entrainment rate curves for all three

models match well the curve of Price [1979]. However, the curves for the mixed layer depth deviate from each other due to differences in the initial phase of the ex- periment after the forcing is suddenly switched on.

It has already been shown by Burchard and Baumert [1995] that c•3 should have negative values for stable stratification, which is in contrast to other findings. For instance, Omstedt et al. [1983] use c•3 - 0.8, and Rodi [1987] suggests values between 0.0 and 0.2 for stable stratifications. The difference is probably due to differ- ences in the stability functions and the values for kmin and Zmin- A negative value for c•3 provides a source for dissipation in the case of stable stratification. This has a similar effect on the lower limit for e, equation (16), which is included into the model in order to parameter- ize the effect of transformation of turbulent eddies into

internal waves.

For unstable stratification, ce3 has to be positive in order to provide a source for • in pure convection sce- narios, that is, in the absense of shear production P. Here the value c•3 - I suggested by Rodi [1987] has been adopted. It should be noted that the shift from a positive to a negative value for c•3 at neutral stratifica- tion (i.e., B - 0) does not introduce any discontinuity into the model.

It has been reported by several authors that differen- tial models like the k-e and MY models used here cause

a deepening of the mixed layer which is too slow [see Martin, 1985; Kantha and Clayson, 1994]. The inclu- sion of the very simple internal wave model by prescrib- ing lower limits for k and e somehow accelerates the

10,550 BURCHARD ET AL.' TWO-EQUATION TURBULENCE MODELS

0

-5

-10

E -15

N -20

-25

-30

_

/

/

0 1 2 3 4 0E+0

L/m 0E+0 2E-4 4E-4 6E-4 1E-2

k / (m2s -2) v, t / (m2s-1) 2E-2

Figure 5. Simulation of wind-induced mixed layer deepening: Profiles of (left) turbulent kinetic energy, (middle) macro length scale, and (right) eddy diffusivity after 30 hours of surface stress. Read 2E-4 as 2 x 10 -4. Dashed line, constant k-e model; thin line, quasi-equilibrium k-e model; thick line, quasi-equilibrium MY model.

mixed layer deepening in such a way that it matches the theory of Price [1979]. This can of course also be achieved by the more complex internal wave models as suggested by Mellor [1989] and Kantha and Clayson [1994]. However, the discussion of such modeling fea- tures is beyond the scope of this paper as they can be easily added to both k-e and MY models.

An inspection of the vertical profiles after 30 hours shows interesting features (see Figure 5). The profile of turbulent kinetic energy using the constant k-e model shows a nearly linear decrease from the surface down to the mixed layer depth. In contrast, the profile re- sulting from the quasi-equilibrium k-e and the quasi- equilibrium MY model has its maximum at 20 m below the surface. The strong dependency on the choice of the stability function can also be seen for the macro length scale L. In contrast, however, the profiles of eddy dif- fusivity are similar for all models. It can be concluded therefore that the performance of a turbulence model depends critically on the choice of the stability func- tion.

5.3. Monin-Obukhov Similarity

The Monin-Obukhov similarity theory relates the flux- es of velocity and buoyancy in a steady state boundary layer to their gradients by means of a length scale de- fined as

3

L• = - n-•' (34) Gradients of momentum and temperature can then be expressed by

Ozu - --•M (35)

Ozb - -•'•, . (36) Businger et al. [1971] determined the similarity func-

tions •M and •H with the aid of a large experimental data set (see Figure 6)

•M • -- I + 4.7œ• /_.,• >_ O,

(I)M ( z' __ __ __ Z-•) (1 15•-•ff) -1/4 z' LM < O, (37)

3oooooooooo0•i$•©•

-0.2 -0.1 0.0 0.1 0.2

0

-0.2

0 Constant k-œ model •]• Quasi-equilibrium k-œ model • Quasi-equilibrium MY model

Curve of Businger et el. [1971] ...... Mellor-Yarnada [1982] zorn eq. model

-0.1 0.0 0.1 0.2

z/L a

Figure 6. Mønin-Obukhøv similarity functions (I)M and (I'H versus the ratio of surface distance and Monin- Obukhov length scale. The symbols denote model re- sults; the thin line shows the Businger et al. [1971] em- pirical relations, and the dashed line shows the zero- equation Mellor and Yamada [1982] algebraic relations.

BURCHARD ET AL.: TWO-EQUATION TURBULENCE MODELS 10,551

L• >0, (38)

( ß n • -0.74 1-9Z--• •

10,552 BURCHARD ET AL.: TWO-EQUATION TURBULENCE MODELS

6. Stratified Tidal Flow

The classical test cases for evaluating mixed layer models like the Ocean Weather Ship station Papa in the Pacific [see Martin, 1985; Kantha and Clayson, 1994] and the Fladenground Experiment data [see Friedrich, 1983; Frey, 1991; Burchard and Baumert, 1995] do not include turbulence measurements. In this case we use

the Irish Sea measurements of Simpson et al. [1996], which include observation of the dissipation of turbu- lent kinetic energy. The measurements were made over a 24 hour period in July 1993 at a site with water depth of 90 rn and position 53ø49'N, 5ø27'W. The site has rectilinear tidal currents, and the tidal current ampli- tude at the time of the measurements was 0.45 rn s -z.

During the observational period the water column was thermally stratified with near surface temperature of 14øC and temperatures of 10øC below the thermocline, which was situated 75 rn above the seabed. In addition

to hourly conductivity-temperature-depth profiles and moored current meter measurements, turbulent dissi- pation was measured using a FLY free-fall probe. Six profiles were made an hour, and these were averaged together to give hourly means. Measurements made in the top 10 rn of the water column have been excluded as the probe may still be accelerating, and the signal may be contaminated by the ships wake.

The forcing data for the model is provided by a time series of current measurements taken 12 rn above the

bed, from which a sea surface slope is calculated so that the model exactly reproduces the observed flow. Cross- surface heat exchange is calculated from dew point tem- perature, wind speed, and direct solar radiation ob- served at a nearby meterological station. Cross-surface momentum exchange is calculated from the observed wind speed and direction [following Simpson et al.,

Figure 7 shows isopleth diagrams of measured and modeled dissipation rate. The model results are shown for quasi-equilibrium stability functions and reproduce the observed M4 period oscillations in the bottom layer (caused by the flooding and ebbing of the predominant semi-diurnal M2 tide). However, the variability of the dissipation rate in the stratified core region of the flow cannot be resolved by the simple internal wave model included.

A major difference between the k-s model and the MY model is evident in the mean profiles of dissipa- tion rate (see Figure 8). The predicted bottom mixed layer depth is greater using the MY models. Other- wise, the quasi-equilibrium version of the k-s and the MY model fit well. Models using both versions of the length scale prescription for Lz, (31) and (32), have been included in Figure 8. The difference between the resulting dissipation profiles is negligible. This is prob- ably because the main difference between the Lz oc- curs in the core regions, which in the present situation are mainly controlled by the stability, while near the

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

Figure 8. Irish Sea simulation: Mean profiles of dis- sipation rate from measurements, It-e, and MY models.

two boundaries both L z forms follow nz'. It should be noted that the good agreement between the measure- ments and the model prediction in the core region of the flow are a consequence of the tuning of kmin to a value of 7.6 x 10 -6 m 2 s -2.

Time series of dissipation rate s, extracted from the measurements and the model results 12 rn above the

bed (see figure 9, the same height as the current mea- surements) shows variation in the bottom mixed layer depth. The top panel in Figure 9 shows hourly means of measured dissipation rate 12 rn above the bed, in comparison with log-law estimates for s

3

slog : U, (39)

where u, is the friction velocity calculated from the measured velocity lul

u, - lul (40) In (d'+z• )

with the bottom roughness length z0 b - 0.0015 rn and a height above the sea bed, ds = 12 m.

The phase lag between current and dissipation rate is about 1 hour (for a systematic investigation on such hysteresis effects see Baumert and Radach [1992]). The reason for this phase lag is that the velocity shear near the bed, and thus the turbulence, have to be trans- ported upwards to the position ds where the dissipation rate and the mean current is measured. This phase lag is reproduced by both models. Figure 9 shows that the k-s model predicts a larger phase lag than the MY model, the difference is about 20 min. The current set of measured dissipation rates is not sufficiently accu- rate to evaluate which of the models is more realistic.

BURCHARD ET AL' TWO-EQUATION TURBULENCE MODELS 10,553

1 E-5

1 E-6

E • 1E-7

1 E-8

1 E-5

• 1E-6

E • 1E-7

1E-8

1 E-5

1 E-6

E • 1E-7

1E-8

--- • Measured (lh means) - Log-law estimate

_

-- k - • model I • Constant stability functions - ! • Quasi-equilibrium stability functions i

-: MY level 2.5 model,

10,554 BURCHARD ET AL.: TWO-EQUATION TURBULENCE MODELS

Galperin, B., L.H. Kantha, S. Hassid, and A. Rosati, A quasi-equilibrium turbulent energy model for geophysical flows, J. Atmos. Sci., J5, 55-62, 1988.

Jobson, H.E., and W.W. Sayre, Vertical mass transfer in open channel flow, J. Hydraul. Div. Am. Soc. Civ. Eng., 96(HY3), 703-724, 1970.

Kantha, L.H., and C.A. Clayson, An improved mixed layer model for geophysical applications, J. Geophys. Res., 99, 25235-25266, 1994.

Karo, H., and O.M. Phillips, On the penetration of a turbu- lent layer into stratified fluid, J. Fluid Mech., 37, 643-655, 1969.

Kraus, E.B., and J.S. Turner, A one-dimensional model of the seasonal thermocline, Tellus, 19, 98-106, 1967.

Large, W.G., and G.B. Crawford, Observations and simula- tions of upper ocean response to wind events during the Ocean Storms Experiment, J. Phys. Oceanogr., 25, 2831- 2852, 1995.

Luyten, P.J., E. Deleersnijder, J. Ozer, and K.G. Ruddick, Presentation of a family of turbulence closure models for stratified shallow water flows and preliminary application to the Rhine outflow region, Cont. Shelf Res., 16, 101-130, 1996a.

Luyten, P.J., J.H. Simpson, and T.P. Rippeth, Comparison of turbulence models for homogeneous and stratified flows with turbulence measurements in the Irish Sea, paper presented at MAST Workshop on Turbulence Modelling, Bergen, Norway, Aug. 8-10, 1996, 1996b.

Martin, P.J., Simulation of the mixed layer at OWS Novem- ber and Papa with several models, J. Geophys. Res., 90, 903-916, 1985.

Mellor, G.L., Retrospect on oceanic boundary layer mod- elling and second moment closure, in Parametrization of Small-Scale Processes; Proceedings of the Aha Hulikoa Hawaiian Winter Workshop, edited by P. Mfiller and D. Henderson, pp. 251-271, Univ. of Hawaii at Manoa, Hon- olulu, 1989.

Mellor, G.L., and T. ¾amada, A hierarchy of turbulence closure models for planetary boundary layers, J. Atmos. Sci., 31, 1791-1806, 1974.

Mellor, G.L., and T. Yamada, Development of a turbu- lence closure model for geophysical fluid problems, Rev. Geophys., 20, 851-875, 1982.

Mohammadi, B., and O. Pironneau, Analysis of the k-z Tur- bulence Model, Masson, Paxis, 1994.

Nakagawa, H., I. Nezu, and H. Ueda, Turbulence in open channel flow over smooth and rough beds, Proc. Jpn. Soc. Civ. Eng., ϥ1, 155-168, 1975.

Omstedt, A., J. Sahlberg, and U. Svensson, Measured and numerically simulated autumn cooling in the Bay of Both- nia, Tellus, 35A, 231-240, 1983.

Patankar, S.V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.

Price, J.F., On the scaling of stress-driven entrainment ex- periments, J. Fluid Mech., 90, 509-529, 1979.

Rodi, W., Turbulence models and their application in hydraulics, report, Int. Assoc. for Hydraul. Res., Delft, Netherlands, 1980.

Rodi, W., Examples of calculation methods for flow and mixing in stratified flows, J. Geophys. Res., 92, 5305- 5328, 1987.

Simpson, J.H., J. Sharples and T.P. Rippeth, A prescrip- tive model of stratification induced by freshwater run-off, Estuarine Coastal Shelf Sci., 33, 23-35, 1991.

Simpson, J.H., W.R. Crawford, T.P. Rippeth, A.R. Camp- bell, and J.V.S. Cheok, The vertical structure of tur- bulent dissipation in shelf seas, J. Phys. Oceanogr., 26, 1579-1590, 1996.

Stelling, G.S., Compact differencing for stratified free sur- face flow, Adv. Hydro-Sci. Eng., 2, 378-386, 1995.

Telbany, M.M.M.E., and A.J. Reynolds, The structure of turbulent plane Couette flow, J. Fluids Eng., 10•, 367- 372, 1982.

Ueda, H., R. MSller, S. Komori, and T. Mizushina, Eddy diffusivity near the free surface of open channel flow, Int. J. Heat Mass Transfer, 20, 1127-1136, 1977.

H. Burchard, Joint Research Centre, Space Applications Institute, TP 690, 1-21020 Ispra (VA), Italy. (e-mail: hans.burchard@jrc.it)

O. Petersen, International Research Centre for Compu- tational Hydrodynamics, Danish Hydraulic Institute, Agern All• 5, DK-2970 H0rsholm, Denmark. (e-mail: osp@dhi.dk)

T. P. Rippeth, School of Ocean Sciences, University of Wales, Bangor, Menai Bridge, Gwynedd, LL59 5EY, Wales. (email: oss009@bangor.ac.uk)

(Received January 31, 1997; revised January 14, 1998; accepted January 21, 1998.)