Direct numerical simulations (DNS) are performed to investigate the evolution of turbulence in a uniformly sheared and stably strati ed flow. The spatial discretization is accomplished by a spectral collocation method, and the solution is advanced in time with a third-order Runge{Kutta scheme. The turbulence evolution is found to depend strongly on at least three parameters: the gradient Richardson number Ri, the initial value of the Taylor microscale Reynolds number Re , and the initial value of the shear number S K= . The e ect of each parameter is individually studied while the remaining parameters are kept constant. The evolution of the turbulent kinetic energy K is found to follow approximately an exponential law. The shear number S K= , whose e ect has not been investigated in previous studies, was found to have a strong non-monotone influence on the turbulence evolution. Larger values of the shear number do not necessarily lead to a larger value of the eventual growth rate of the turbulent kinetic energy. Variation of the Reynolds number Re indicated that the turbulence growth rate tends to become insensitive to Re at the higher end of the Re range studied here. The dependence of the critical Richardson number Ricr, which separates asymptotic growth of the turbulent kinetic energy K from asymptotic decay, on the initial values of the Reynolds number Re and the shear number S K= was also obtained. It was found that the critical Richardson number varied over the range 0:04 < Ricr < 0:17 in our DNS due to its strong dependence on Reynolds and shear numbers.

During the past decade, both numerical simulations and laboratory experiments
of homogeneous turbulence in a stably strati ed shear flow have been performed
to investigate the e ects of shear and strati cation on the turbulence evolution.
These studies were undertaken with the goal of explaining and understanding the
turbulent microstructure of geophysical flows in the atmosphere and ocean. The
present paper considers strati ed shear flow with a uniform gradient of the ambient
density S and uniform mean shear rate S . The primary non-dimensional parameter
that determines the turbulence evolution in such a flow is the gradient Richardson
number Ri = N2=S 2, the square of the ratio of the Brunt{Vašisašlša frequency N =
[(?gd =dz)= 0]1=2 and the mean shear rate S . A critical value of the Richardson
number Ricr can be found for which the turbulent kinetic energy K stays constant
in time, grows in time for Ri < Ricr, and decays in time for Ri > Ricr. Knowledge
of the critical Richardson number is therefore important as it divides the asymptotic
fate of the nonlinearly evolving turbulence into growth or decay. The signi cance of
the Richardson number was rst mentioned in Taylor's 1914 Adams Prize Essayy.
Subsequent studies by

The rst direct numerical simulations (DNS) of homogeneous turbulence in a strati ed shear flow were performed by Gerz, Schumann & Elghobashi (1989). In their investigation shear periodic boundary conditions were used in combination with a nite di erence/spectral collocation method. The simulations were performed on a grid with 643 points. The Richardson number was varied over the range 0 6 Ri 6 1. Gerz et al. found that the evolution of the turbulent kinetic energy K depends strongly on the Richardson number Ri. The initial energy partition , the ratio of the potential to kinetic energy, was found to lead to di erent initial transients but not to influence the turbulence evolution over a long period of time. Gerz et al. investigated the occurrence of counter-gradient heat fluxes which were more prominent for large Richardson numbers and high molecular Prandtl numbers.

Holt, Kose & Ferziger (1992) investigated the turbulence evolution in a
homogeneous stably strati ed shear flow using purely periodic boundary conditions in
combination with a spectral collocation method. The simulations were performed
on a grid with 1283 points. The simulations covered a parameter range 0 6 Ri 6 1
and 20 6 Re 6 100, where Re is the Taylor microscale Reynolds number. They
divided the turbulence evolution into shear- or buoyancy-dominated regimes for the
Richardson number smaller or larger than the transitional value Rit for which the
vertical density flux vanishes. Simulations with constant turbulent kinetic energy K
were found to lie in the shear-dominated regime, that is Ricr < Rit. Holt et al.
investigated the influence of the initial energy partition and of the molecular Prandtl
number P r and agreed with the conclusions of

Piccirillo & Van Atta (1997) investigated the turbulence evolution in a homogeneous
stably strati ed shear flow using a thermally strati ed wind tunnel, in which they
varied the Reynolds number Re over a small range by using a variety of di erent
turbulence-generating grids. They found a decrease of the critical Richardson number
Ricr with increasing grid size and thus increasing Reynolds number Re .

The evolution of high-shear-number flow has been studied analytically and
numerically. For high shear numbers S K= , the nonlinear term in the momentum equation
may become less important than the shear forcing term. Therefore the nonlinear term
can be neglected, and a simpli ed analysis can be performed instead. This approach
is known as rapid distortion theory (RDT). An introduction to RDT is given in the
review article by

A motivation of this study is to explain the apparently di erent dependence
of the critical Richardson number Ricr on the Reynolds number Re observed in
direct numerical simulations by

An additional motivation of the present study is to perform direct numerical simulations at the higher Reynolds numbers accessible with today's supercomputers, in order to investigate the possibility of a decreased influence of the Reynolds number Re on the turbulence evolution at high values of Re .

In the following section, the evolution and transport equations used in this study are introduced, and the non-dimensional parameters governing the turbulence evolution are derived. In x3 the numerical method is described, and the initial conditions and their importance for e ective parameterization are addressed. The dependence of the evolution of the turbulent kinetic energy on the non-dimensional parameters derived in x2 is presented in x4. In x5 the asymptotic evolution of the turbulent kinetic energy is summarized. The dependence of the critical Richardson number on the remaining non-dimensional parameters is discussed in x6. Section 7 summarizes our conclusions on the turbulence evolution in a stably strati ed shear flow. The numerical method used in this study is validated in an Appendix.

In this section the equations of motion used to describe the evolution of a turbulent strati ed shear flow are presented. In addition the transport equations for secondorder moments are introduced. Finally the non-dimensional parameters governing the turbulence evolution are derived.

Our study of a turbulent strati ed shear flow is based on the continuity equation of an incompressible fluid, the Navier{Stokes equation and a transport equation for the density. In the following, Xi is the ith coordinate of an orthonormal Cartesian coordinate system, Ui is the ith component of the total velocity, % is the total density, P is the total pressure, g is the gravitational constant, is the molecular viscosity of the fluid, and is the di usion coe cient. The dependent variables Ui = Ui + ui, % = % + , and P = P + p are decomposed into a mean part (denoted by an overbar) and a fluctuating part (denoted by lower-case or alternative greek letters). The mean velocity Ui = S X3 i1 and the mean density % = 0 + S X3 are given by the constant velocity gradient S = @U1=@X3 and the constant density gradient S = @%=@X3, as shown in gure 1. It is assumed that a mean pressure gradient balances the mean buoyancy force, that is 0 = ?@P =@X3 ? g( 0 + S X3). This decomposition is introduced into the equations of motion. Furthermore the Boussinesq approximation is employed.

Spectral accuracy in the spatial discretization can be obtained by the use of periodic
boundary conditions, but due to the e ect of shear, periodic boundary conditions
cannot be applied directly to the equations of motion. In previous simulations,

Turbulence evolution in a uniformly sheared and stably strati ed flow 1 @p @p 0 @xi ? S t @x1 i3 ?G i3 + ;

1 ReP r

Here Re = UL= is the Reynolds number, P r = = is the Prandtl number, and G = gL=U2 is the non-dimensional gravity coe cient. U, L, and 0 are characteristic scales for velocity, length, and density, respectively. The numerical method used to solve this set of equations is introduced in x3.1.

In this subsection the transport equations for second-order moments are derived.
The overbar a denotes the volume average of a, which is the appropriate Reynolds
average in the case of homogeneous flow studied here. The homogeneity is preserved
due to the uniformity of the ambient density gradient S and the mean shear rate S
as discussed by

(ui j3 + uj i3); Tij =

; B =

u3 ; g 0 =

Here Pij is the production term, Bij the buoyancy term, Tij the pressure-strain term, and ij the dissipation term. The trace of this equation gives the transport equation for the kinetic energy K = uiui=2: d dt

d

K = dt 21 uiui = P ? B ? ;

In addition, the anisotropy tensor bij is de ned as

The transport equations for the mass fluxes Mi = ui are derived from the momentum equation and the transport equation for the density: d dt

Mi = ?S ui ? S uiu3 ? i3 + :

Finally the transport equation for the density variance transport equation: is derived from the (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) @xk @xk The potential energy K can be computed from the density variance: 2.3. Parameters governing the turbulence evolution To derive the non-dimensional parameters governing the turbulence evolution, the transport equation for the turbulent kinetic energy K is scaled. Let q be the characteristic velocity scale, l the characteristic length scale and r the characteristic density scale. A characteristic time scale for the turbulence evolution is then given by = l=q. In addition the Taylor microscale is introduced as a derivative length scale by = 5 q2= 2. The terms of the transport equation for the kinetic energy K are divided by the dissipation : and scaled as follows: 1 d

dt | {1z }

K =

P |{z} 2 ?

B |{z} 3

?1 O(term 1) = Re =l;

O(term 2) = S K= ; O(term 3) = Ri Le l

S K 2 1 l

Re

: g 0 K = 1 g 2 0jS j :

Here Re = q = denotes the Reynolds number based on the Taylor microscale, Ri = ?(gS )=( 0S 2) the Richardson number, and Le = r=S the Ellison scale. The kinetic energy is K = O(q2). Therefore, the non-dimensional parameters governing the turbulence evolution for this low-Reynolds-number scaling are the Richardson number Ri, the Taylor microscale Reynolds number Re and the shear number S K= . In addition, the initial conditions influence the turbulence evolution by the length-scale ratios =l of the velocity elds and Le=l of the density eld.

For high Reynolds numbers, the dissipation varies as = O(q3=l) or equivalently O(l= ) = Re . This simpli es the scaling to

O(term 1) = 1;

O(term 2) = S K= ; Turbulence evolution in a uniformly sheared and stably strati ed flow 2 O(term 3) = Ri Le S K : (2.25) l

Therefore, the non-dimensional parameters governing the turbulence evolution for the high-Reynolds-number scaling are the Richardson number Ri, the shear number S K= , and the ratio Le=l.

Additional parameters can be obtained from the evolution equation of the potential
energy. These parameters are not considered in this investigation. E ects of a variation
of the molecular Prandtl number P r and the initial energy partition , the ratio of
potential to kinetic energy, were considered by

We emphasize that, for a given initial spectral shape of the flow perturbations, a parameterization based on the initial values of the Reynolds number Re and the shear number S K= is possible, because initially di erent values of these parameters remain di erent throughout all simulations performed in this study, as shown in gures 17(a) and 17(b).

In this section the numerical method is described, and the initial conditions and their importance are discussed. A validation of the numerical method is given in the Appendix.

The numerical scheme uses a spectral collocation method for the spatial discretization. To compute spatial partial derivatives, the dependent variables are transformed into Fourier space using the fast Fourier transformation algorithm, multiplied with the corresponding wavenumbers, and transformed back into physical space. Second derivatives are computed by applying this method successively. The nonlinear terms are computed in physical space.

The solution is advanced in time using a third-order Runge{Kutta method. During
the time advancement the coordinate system in the moving frame of reference becomes
more and more skewed. Following a method originally devised by

The results presented here are from simulations using 1283 grid points, with the following exceptions. The low-Reynolds-number Re 6 22:36 and low-shear-number S K= 6 4:0 simulations are performed on a 963 grid, and the high-Reynolds-number simulations Re > 67:08 are performed on a 1443 grid.

The initial spectra have to be de ned carefully to allow a parameterization of the flow evolution based on the initial values of the parameters discussed in x2.3. Usually a random number generator is used to produce the initial fluctuating elds. These elds follow a speci ed initial energy spectrum and ful l the continuity equation, but they contain no phase information and have no initial spectral transfer. The energy spectrum E is half the shell average of the squared amplitude of the threedimensional Fourier transformation of the velocity. Initial energy spectra used in K K0 1.0 0.8 0.6 0.4 0.2 0 2 4

6
St
8
10
previous investigations include the top-hat spectrum

E(k) =

E0 for k1 < k < k2 0 elsewhere,

E(k) = E0k4 exp(?2k2=k02):

Unfortunately these spectra result in an initial transient in the simulations that does not allow a parameterization based on initial values. In gure 2 the evolution of the turbulent kinetic energy K is shown for simulations with di erent initial radial spectra but identical initial values of the Richardson number Ri = 0:08, the Reynolds number Re = 22:36, and the shear number S K= = 2:0. The simulations with equations (3.1) and (3.2) as initial spectra show a large initial decay of K during the transient phase and nally continue to decay. In gures 3(a) and 3(b) the evolution of the Reynolds number Re and the shear number S K= are shown. During the initial transient these numbers exhibit a large drop. Therefore their initial values are not characteristic of the asymptotic evolution of the turbulence.

To reduce the initial drop in K, Re , and S K= , an alternative initialization method
is introduced. The actual simulation of a strati ed shear flow uses as initial elds
the resulting elds of a simulation of unstrati ed unsheared isotropic turbulence.
The initial value of the Reynolds number of the initialization simulation is chosen
such that the nal value matches the target initial Reynolds number of the actual
simulation. The initialization simulation is advanced for about one eddy-turnover
time and thus beyond the initial transient. During this time the skewness of @u=@x,
which is a measure of the spectral transport, increases from zero to a nal value
S k = ?0:46. This value is in good agreement with other experimental and numerical
results of ?0:5 < S k < ?0:4

E(k) = E0k2 exp(?2k=k0): The results of simulations started from these initial conditions are also shown in (3.1) (3.2) (3.3) Rek

SK e 40 30 20 10

0 8 6 4 2 0

Isotropic 2 2 ~k2exp(?2k/k0)

~k4exp(?2k2/k20) 4 4 ~k4exp(?2k2/k20)

St St

Top hat 6 6

Isotropic

gures 2, 3(a) and 3(b). They result only in a small initial decay of the turbulent kinetic energy K, which has a physical explanation that will be discussed in x4. Neither Re nor S K= show an initial drop. The simulations initialized with spectra (3.2) and (3.3) use a peak wavenumber k0 = 8. The spectrum of the isotropic turbulence has a peak wavenumber k0 = 7. The simulation initialized with the top-hat spectrum uses the wavenumbers k1 = 6 and k2 = 12.

During the initial transient, the energy spectrum de ned by the initial conditions evolves into a spectrum characteristic for shear flow. Figure 4(a) shows the evolution of the spectrum for the simulation started from isotropic initial conditions. In the initial phase of the simulation (0 < S t < 5), the low-wavenumber portion gains energy, and the high-wavenumber portion loses energy, but the general shape of the spectrum changes only slightly. A very similar evolution, though not shown, was observed for the spectrum of the simulation initialized with spectrum (3.3). Figure 4(b) shows the evolution of the spectrum for the simulation initialized with 20 k 20 k

St = 5

St = 5

St = 9 30

St = 0

40 30 40 the top-hat spectrum. Comparison shows that the initial state does not contain a su cient amount of energy at the lowest wavenumbers and in the high-wavenumber portion of the spectrum. During the initial transient, energy is redistributed into these wavenumber portions. This increases strongly the viscous dissipation , which in turn results in the large drop of S K= and Re . This renders the initial values of these parameters uncharacteristic for the prediction of the asymptotic state of the flow. A similar evolution was observed for the spectrum of the simulation initialized with spectrum (3.2).

The initial condition of well-developed isotropic turbulence from a previous direct numerical simulation was chosen, because it represents a solution of the equations of motion for unstrati ed unsheared flow, and it allows an e ective parameterization based on initial values. Isotropic initial conditions also compare better with the initial conditions found in experiments, where nearly isotropic conditions develop behind the turbulence-generating grid before the e ects of shear and strati cation become

Ri important. All the shear flow simulations presented in this paper are based on the initialization method using well-developed isotropic turbulence elds obtained from a separate isotropic turbulence simulation initialized with spectrum (3.3).

This section presents the results of a series of simulations, in which the Richardson number Ri, the initial value of the Reynolds number Re , and the initial value of the shear number S K= are varied independently. Each section describes the influence of one parameter on the turbulent kinetic energy evolution and gives a physical explanation for that influence.

Table 1 gives an overview of the simulations described in this section. All simulations are initialized with velocity elds taken from simulations of unstrati ed unsheared decaying isotropic turbulence with no density fluctuations. The molecular Prandtl number P r = 0:72 is xed.

All simulations show qualitatively the same dependence on Ri. Therefore one set of simulations with the initial values Re = 44:72 and S K= = 2:0 is discussed here.

Figure 5(a) shows the evolution of the turbulent kinetic energy K as a function of the non-dimensional time S t with Ri as the variable parameter. After an initial decay K grows for small Richardson numbers and decays for high Richardson numbers. This makes it possible to de ne a critical Richardson number Ricr for the case of constant K: d dt

K ( > 0 for Ri < Ricr = 0 for Ri = Ricr < 0 for Ri > Ricr.

The initial decay in all cases is due to the isotropic initial conditions that cause
the production term P = ?S u1u3 to be initially zero. During the initial phase the
production term grows. The initial decay has been observed in other numerical
simulations

In the case of homogeneous turbulent unstrati ed shear flow, experiments

Here b13 is the 1{3 component of the shear stress anisotropy tensor bij = uiuj=ukuk ? ij=3. Under the assumption that each term on the right-hand side of equation (4.2) evolves to an asymptotically constant value for large non-dimensional time S t, the equation can be integrated to obtain

K = K0 exp (?S t): (4.3)

The exponential approximation is also shown with dashed lines in gure 5(a). The constant of integration is used to t the graphs. The agreement shows that exponential growth or decay of the turbulent kinetic energy K is a good approximation. The exponential decay of K for the case Ri > Ricr is di erent from the power law decay observed in unsheared decaying initially isotropic turbulence behind a grid. Figure 5(b) shows the evolution of the growth rate ?. Note that a positive value of ? is associated with growth and a negative value with decay of the turbulent kinetic energy K. The asymptotic value of the growth rate ? is positive for small Ri and negative for large Ri. The evolution of the anisotropy b13 is shown in gure 6. The magnitude of b13 decreases with increasing Ri. Therefore the e ect of strati cation reduces the anisotropy of the flow introduced by the e ect of shear. Figure 7(a) shows the evolution of =P for various Ri. The asymptotic value of =P increases with increasing Ri. The evolution of B=P is presented in gure 7(b). The asymptotic value of the ratio B=P increases with increasing Ri.

The value of the critical Richardson number for this set of simulations is about
Ricr = 0:167. (The method used to obtain the value of Ricr is described in x6.)
While the other sets show qualitatively the same Richardson number dependence, the
value of Ricr varies when the initial values of parameters such as Re and the S K=
are changed. The previous numerical simulations

To investigate the influence of shear and buoyancy, the turbulent kinetic energy
equation is rewritten in a non-dimensional form:
Figures 8(a) and 8(b) show the evolution of B= and P = , respectively. As expected
B= grows with the Richardson number. But this increase is too small to account
for the change in the evolution of K. On the other hand P = decreases strongly
with increasing Richardson number. Therefore the primary e ect of buoyancy is
not a direct sink for K through the buoyancy flux but is the indirect reduction of
the shear-induced production of K. This result is in agreement with the previous
investigations by

In this subsection the results from simulations with a xed Richardson number Ri = 0:08 and a xed initial value of the shear number S K= = 2:0 are presented. The initial value of the Taylor microscale Reynolds number Re is varied within the range accessible in this investigation to study its influence on the turbulence evolution.

Figure 9(a) shows the evolution of the turbulent kinetic energy K. Initially K decays due to the isotropic initial conditions. This initial decay increases with increasing Re . After this initial phase K continues to decay for small Re but starts to grow for larger Re .

The evolution of the exponential growth rate ? is shown in gure 9(b). For the low-Reynolds-number simulation with Re = 11:18, the growth rate ? is negative, and the turbulent kinetic energy K decays. For a nite range of moderate Reynolds numbers with Re 6 44:72, the growth rate ? increases with increasing Re , becomes positive, and reaches a maximum. For even higher Reynolds numbers Re > 44:72, the growth rate ? decreases slightly with increasing Re and appears to become relatively insensitive to Re .

A necessary requirement for the Reynolds number independence of the growth rate
? is that the turbulent dissipation rate varies according to the high-Reynolds-number
scaling = Au3=l where A is a constant of order 1. After examining experimental
data on grid turbulence,

The evolution of the anisotropy b13 is presented in gure 10 for various initial e P B P 1.5 1.0 0.5

0 0.5 0.4 0.3 0.2 0.1 0 5 0 10 St 10 St

Ri = 0.20 0.16 15 20 20 values of Re . The asymptotic value of b13 decreases with increasing Re and nally reaches a constant value independent of Re . Figure 11(a) shows the evolution of =P for various Re . The asymptotic value of =P decreases with increasing Re for Re 6 44:72. For Re > 44:72 the ratio =P increases with increasing Re but nally becomes independent of Re . From equation (4.2) it is clear that the minimum of =P at Re = 44:72 causes the maximum of the growth rate ? at this Reynolds number. The evolution of B=P is shown in gure 11(b). The asymptotic value of the ratio B=P becomes approximately independent of the Reynolds number for Re > 44:72.

In this subsection the dependence of the turbulence evolution on the initial value of the shear number S K= is addressed. The Richardson number Ri = 0:06 and the initial value of the Reynolds number Re = 22:36 are xed. P e 0.4 0.3 0.1

0

In gure 12(a) the evolution of the turbulent kinetic energy K is presented as a function of S K= . Initially K decays due to the isotropic initial conditions. This decay is stronger for the simulations with a smaller S K= , because for these simulations the turbulence time scale K= is small compared to the shear time scale 1=S . Therefore the turbulence has more time to decay before the turbulence production due to shear becomes signi cant. The growth rate ? of the turbulent kinetic energy is shown in gure 12(b).

The further evolution of the turbulent kinetic energy K can be divided into three regimes. First, for the low-shear-number simulation with S K= = 0:2, the growth rate ? is negative, and K continues to decay. For this case the turbulence production due to shear P is always smaller than the turbulence destruction due to buoyancy e ects B and viscous dissipation , that is P < B + . Second, for a nite regime of moderate shear numbers 0:5 6 S K= 6 6:0, the growth rate ? is positive, and K grows. The turbulence production is larger than the turbulence destruction, that is P > B + . Third, K K0 1.5 1.0 0.5

0

89.44 89.44 for large shear numbers S K= > 6:0, the growth rate ? is negative, and K decays again. This decay is due to a strongly reduced turbulence production as will be shown below.

Although the decay of the turbulent kinetic energy K for high shear numbers seems
counterintuitive, there is an explanation for this e ect. Consider the unstrati ed case
rst. For high S K= , linear e ects dominate the turbulence evolution, and rapid
distortion theory (RDT) applies at least for short times. It is known, see for example

Now consider the stably strati ed case. For the initial shear number S K= to
have an influence on the evolution of the turbulent kinetic energy K, it is enough
that the stabilizing rapid distortion e ect persists for a time that is long enough
for strati cation e ects to become important. For Ri = 0:06, the interval S t = 12
in the DNS of

The shear number range for which the turbulence asymptotically grows decreases with increasing Richardson number. For su ciently large Richardson numbers the range of growth disappears, and the turbulent kinetic energy K always decays.

In gure 13 the evolution of the anisotropy b13 is shown. The asymptotic value of the magnitude of b13 decreases with increasing S K= . Figure 14(a) shows the evolution of =P . The asymptotic value of =P decreases with S K= for S K= 6 4:0, reaches a minimum, and nally increases with S K= for S K= > 4:0. The minimum of =P at S K= = 4:0 corresponds to a maximum of the growth rate ? at this shear number. The evolution of B=P is presented in gure 14(b). The asymptotic value of B=P is nearly independent of S K= .

The stabilizing e ect of large initial values of S K= is associated with, rst, a decrease of the asymptotic values of b13 as shown in gure 13 and, second, an increase of =P . The rst e ect is on the large-scale, energy-containing eddies that appears in rapid distortion analysis too. The second e ect on =P = 1=(S K= ) 1=(?2b13) is primarily due to the decrease of the asymptotic value of b13. Although the nal values of S K= are larger for larger initial values of S K= , the net e ect is an increase of =P . 2.0 1.5 e 1.0 P 0.5

0

In this section the dependence of the asymptotic values of the exponential growth rate ? on Ri and the initial values of Re and S K= is summarized. The asymptotic values of the growth rate ? are computed as the average of the right-hand side of equation (4.2) for non-dimensional times S t > 8.

Figure 15 shows the dependence of ? on Ri. The initial values of the Reynolds number Re = 44:72 and the shear number S K= = 2:0 are xed. The growth rate ? decreases approximately linearly with increasing Ri.

The variation of ? with the initial value of Re is presented in gure 16(a). The Richardson number Ri = 0:08 and the initial value of the shear number S K= = 2:0 are xed. For low Reynolds numbers, ? increases with increasing Re . The growth rate ? reaches a maximum and decreases slightly. For high Reynolds numbers, ? tends to become independent of Re . 0.2 0.1 c 0 ?0.1 0 0.2 10 10

St St 0.2 1 0.5 20

0.5 8 20

SK/e =2 6 8 4

10

The dependence of ? on the initial value of S K= is shown in gure 16(b). The Richardson number Ri = 0:06 and the initial value of the Reynolds number Re = 22:36 are kept constant. For low shear numbers, the growth rate ? increases with increasing S K= . The growth rate ? reaches a maximum and decreases with a further increase of S K= . Note that only a variation of S K= can lead to two critical cases with ? = 0.

The evolution of Re for the series of simulations with di erent initial values of Re is shown in gure 17(a). The Richardson number Ri = 0:08 and the initial value of the shear number S K= = 2:0 are xed. A parameterization of the exponential growth rate in terms of the initial value of Re is meaningful, because the curves of the Reynolds number evolution do not intersect. In addition, the shape of the growth rate dependence on the Reynolds number, as shown in gure 16(a), does not change substantially with time.

Figure 17(b) shows the evolution of S K= for the series of simulations with di erent 0.2 St initial values of S K= . The Richardson number Ri = 0:06 and the initial value of the Reynolds number Re = 22:36 are xed. The variation of ? can be presented in terms of the initial value of the shear number S K= , because the curves of the shear number evolution do not intersect. Also, the shape of the growth rate dependence on the shear number, as shown in gure 16(b), does not change substantially in time.

The critical Richardson number Ricr is de ned as the value of Ri for which the turbulent kinetic energy K stays constant in time. In x2.3 it was shown that the evolution of the turbulent kinetic energy K depends on Ri and the initial values of Re and S K= . Therefore Ricr is a function of the initial values of Re and S K= .

For the critical case, the turbulent kinetic energy equation (2.9) simpli es to or The coe cients are The low-Reynolds-number scaling equations (2.21) and (2.22) for P = and B= lead to

1 1 1

Ricr = 5l Re S K= ? S K= : (6.3) The high-Reynolds-number scaling equations (2.24) and (2.25) for P = and B= lead to ? 1:

? 1 = 0 Ricr = 1 1

S K=

1 ? S K=

: e P 0.20 0.15 0.05

0 B P 0.10 (a) (b) 0.2 10 10

St St 8 6 4

1

SK/e =1

This dependence of Ricr on Re and S K= is exact for the instantaneous values of Re , S K= , , and . In order to estimate the dependence of Ricr on the initial values of Re and S K= we assume that, rst, the coe cients and are constant and, second, the nal values of Re and S K= are monotone increasing functions of the initial values of Re and S K= . The second assumption is a result of our direct numerical simulations, and the rst assumption is a reasonable approximation to the turbulence state after an initial transient according to our direct numerical simulations.

Then, it can be expected from equation (6.3) that for low Reynolds numbers, Ricr grows linearly with increasing initial values of Re and, from equation (6.4), that for high Re the critical Richardson number Ricr becomes independent of Re .

In addition, it can be expected that Ricr varies with the initial value of S K= as c given by equations (6.3) and (6.4). The shear number dependence in these equations shows that for small S K= , the critical Richardson number Ricr grows with increasing S K= , reaches a maximum, and nally decreases with a further increase of S K= . It is therefore possible to obtain the same value of Ricr for two di erent values of S K= .

Also note that the critical Richardson number can be negative for su ciently small
initial values of the Reynolds and shear numbers. In this case an unstable density
strati cation must be maintained to allow for a constant level of turbulence. The
analysis presented here was inspired by an argument given in

Simulations were performed to nd the dependence of the critical Richardson number Ricr on the initial values of the Reynolds number Re and the shear number S K= numerically. In x5 it was shown that the exponential growth rate ? is approximately a linear function of the Richardson number Ri. Therefore the value of Ricr was determined by identifying two nearby Richardson numbers that cause growth and decay and using linear interpolation.

Figure 18(a) shows the variation of Ricr as a function of the initial value of Re as determined by DNS. The initial value of S K= is kept constant in this series of simulations. The value of Ricr increases with the initial value of Re . For large Re , Ricr varies only slightly with Re .

The dependence of Ricr on the initial value of S K= is shown in gure 18(b). In this series of simulations the initial value of Re is kept constant. With increasing S K= , the critical Richardson number Ricr increases, reaches a maximum, and nally decreases. Therefore the same value of Ricr can be obtained at two di erent S K= .

Thus, the variation of the critical Richardson number Ricr as determined from DNS agrees qualitatively with the variation indicated by the scaling analysis of the transport equation for the turbulent kinetic energy.

The laboratory experiments of Piccirillo & Van Atta (1997) showed that when the
mesh size (and thereby the microscale Reynolds number) of the grid was increased,
Ricr decreased, in apparent contradiction to the nding of

0 c 0.05

0 ?0.05 0 20 40 60 80

100 grid increases. The shear number values in the laboratory were below 0:5, placing them in the left-hand corner of the plot in gure 18(b) and thus in the range where Ricr decreases with decreasing S K= . Thus, the decrease in Ricr with increasing mesh size in the laboratory experiment may be a shear number e ect and not a Reynolds number e ect.

In this investigation, a spectral collocation method for the direct numerical simulation of homogeneous turbulence in a strati ed shear flow was implemented and validated. In addition, the importance of carefully de ned initial conditions for an e ective parameterization of the turbulence evolution was addressed. A dimensional analysis was performed to derive the non-dimensional parameters governing the temRek SeK 10 100 50 0 20 15 5 0 0.2 SK/e = 6

8 10 poral development of the turbulence. These parameters are the Richardson number Ri, the Reynolds number Re , and the shear number S K= . From the simulations, it was found that, for a given spectral shape of the initial fluctuations, the evolution of the turbulent kinetic energy K depends strongly on Ri and the initial values of Re and S K= . Furthermore, it was shown that the evolution of K follows approximately an exponential law.

The variation of the asymptotic value of the exponential growth rate ? with Ri, the initial value of Re , and the initial value of S K= is shown in gures 15, 16(a), and 16(b), respectively. The growth rate ? decreases approximately linearly with increasing Ri. The stabilizing influence of the Richardson number on the turbulent kinetic energy is not caused by the explicit sink associated with the buoyancy flux. It Ricr Ricr 0.08 is due to the decreasing value of the relative turbulence production P = as well as the decreasing magnitude of Reynolds shear stress anisotropy b13 with increasing Ri.

The growth rate ? increases with increasing initial values of Re but nally tends to become independent for high Re .

The dependence of ? on the initial value of S K= is non-monotone. The growth rate ? increases, reaches a maximum, and nally decreases with increasing S K= . The decrease of ? as a function of S K= for large shear numbers is consistent with linear viscous RDT results. Although RDT is strictly applicable for short times only, DNS shows that the qualitative e ect of the shear number suggested by RDT persists for long times too. The stabilizing e ect of increased shear number is associated mainly with a decreased Reynolds shear stress anisotropy b13 and also has a contribution from a reduced relative production P = . The shear number e ect discussed here is based on simulations at an initial Reynolds number Re = 22:36 ( nal values as large as Re ' 50) and a Richardson number Ri = 0:06. It is probable that the detailed influence of the shear number may vary with the initial value of the Reynolds number and the Richardson number and should be the subject of a future study.

The critical Richardson number Ricr is of interest, because for Ri < Ricr the
asymptotic fate of the turbulence is growth as opposed to decay for Ri > Ricr. The
value of Ricr was found to depend on the initial values of Re and S K= . The
variation of Ricr with Re and S K= is shown in gures 18(a) and 18(b), respectively.
The variation of Ricr in the DNS is large, ranging from 0:04 at the lowest values of
Reynolds and shear number to a high of 0:17. Thus in nonlinearly evolving uniform
shear flow, the estimate of Ricr = 0:25 motivated by the linear result of

The authors acknowledge the support from the O ce of Naval Research, Physical Oceanography Program, through the grant ONR N00014-94-1-0223. F. G. Jacobitz was partially supported by the Deutschen Akademischen Austauschdienst (DAADDoktorandenstipendium aus Mitteln des zweiten Hochschulsonderprogramms). Supercomputer time was provided by the San Diego Supercomputer Center (SDSC) and the US Army Corps of Engineers Waterways Experiment Station (WES).

To ensure the reliability of the numerical method used in this study, a number
of validation simulations were performed. First, the solutions were compared with
results obtained by

A simulation was performed that matches the initial radial spectrum and the initial
values of the non-dimensional parameters of the simulation labelled HD by

St St 10 10 A simulation with the initial parameters Ri = 0:08, Re = 67:08, and S K= = 2:0 originally performed on a grid with 1283 points was restarted at S t = 9:0 on a grid with 1443 points. The extrapolation necessary for this grid expansion was done in spectral space. Figure 20(a) shows the two-point correlations R11(x) at the nondimensional time S t = 14 of the two simulations performed on the 1443 and 1283 grids. Both cases show that R11(x) decays to zero for su ciently large x. Therefore the resolution of the computational domain does not influence the evolution of the largest turbulence scales.

To ensure a su cient resolution of the small turbulence scales the energy spectra E(k) at the non-dimensional time S t = 14 of the two simulations performed on the 1443 and 1283 grids are shown in gure 20(b). The spectra show an identical evolution R11 2 x 40 k 3 60 1283 1443 of the low-wavenumber portion and a su cient resolution of the high-wavenumber portion of the spectra. Therefore the grid used has the high resolution required to obtain accurately the evolution of the turbulence.

Universality of the non-dimensional parameters The universality of the non-dimensional parameters was checked by rescaling the initial conditions of a simulation with the initial parameters Ri = 0:08, Re = 67:08, and S K= = 2:0. The initial velocity fluctuations q are increased to a new value q = 2q. To keep the Reynolds number constant for a xed initial spectrum, the kinematic viscosity has to be changed to a new value = 2 . To keep the shear number S K= constant, the shear rate S has to be changed to a new value S = 2S . The ratio of the turbulent kinetic energies K =K obtained from the rescaled and K * K original simulations is shown in gure 21. This ratio always remains very close to its initial value of 4.0 as it should. This shows the universality of the non-dimensional parameters derived in x2.3.