Fluctuations, Reynolds Stresses, and Eddy Coefficients

Hydrodynamic Equations of the Sea

3.12 Fluctuations, Reynolds Stresses, and Eddy Coefficients

Because the depth of the ocean is such a small fraction of the radius of the earth (0.08%), a useful approximation is to assume that r in these equations is simply R e , and that the radial derivatives can be replaced by l)/&,.

The full spherical coordinate system must be used, rather than the beta plane approximation, whenever the north-south scale of the motion is large enough for variations in the numerical value of

6

to be important; another instance is when the azimuthal connectedness of the motion plays a role, as is the case for the Antarctic Circumpolar Current.

Hydrodynamic Equations of the Sea 111

Log frequency

Fig. 3.18 Three types of energy spectra for oceanic motions. (a) Solid line: typical

“red” spectrum showing monotonic increase toward lower frequencies up to the spectral maximum at f l o w . (b) Dashed line: spectrum showing a gap at fgap, which divides slow from fast motions. (c) Dotted line: spectrum showing a peak response at fpeak, characterizing narrowband processes such as tides and swell.

tral gap and is one for which an averaging over time scales shorter than l/f, would not seriously distort the description of the dynamics at lower frequencies. If such a gap can be shown to exist even approximately, than the procedure to be derived below would quite possibly be a valid one.

Consider a series of measurements of the generic fluid quantity, y(x,t), made at some point, x, as a function of time. It is altogether possible, even likely, that another set of measurements made at the same place at another time would yield a quite different time series, because in geophysics (as with other branches of nonlaboratory science), randomness is a highly pervasive characteristic. The question then arises as to how to define the statistical moments of y, i.e., values such as the mean, variance, and higher-order functions of the distribution. Now in statistical mechanics, the idea of ensemble averaging is a central concept, wherein an experiment of interest is repeated a large number of times, N , under presumably identical conditions, during which the macroscopic initial and boundary values of y are maintained es- sentially constant. The ensemble average of y may then be obtained by the operation

(3.98)

where the bra ( 0 and ket (>) notation denotes an ensemble average.

While such repeated experiments may be practicable in the laboratory, they are next to impossible in geophysics (and totally impossible in astrophysics), because one cannot properly prepare the initial state of the natural system prior to each set of observations. Even in a system as controllable as one in the laboratory, it is often found that the detailed development of the fluid field under study is not reproducible from one run t o the next, in spite of carefully prepared initial conditions; it is likely that only the values of the mean and standard deviation of the field are repeatable, while the finer- grained aspects of the motion are not. In fluids, this is often due t o the exis- tence of small-scale instabilities that grow unpredictably out of random fluctuations as the system evolves. Nevertheless, carefully defined ensemble averaging is the preferred method for arriving at the best estimates of the field.

An alternative method of averaging is to average over time, expecting that, if the series of observations is continued for a sufficiently long time, the system will ultimately pass through all of its microstates, with each state occur- ring as frequently as its statistics dictates. The mathematical process analogous to Eq. 3.98 is

(3.99)

where we have denoted the temporal average by an overbar, and the averaging interval by 2t,. If the random geophysical process is statistically station- ary, i.e., the statistical moments of its distribution (mean, variance, kurtosis, etc.) are found to be the same when the experiment is repeated many times over, then the time average is expected to yield the same estimates of y as the ensemble average, and in the sense of numerical equality, one has

The assumption of equivalence of temporal and ensemble averaging is termed the ergodic hypothesis of statistical mechanics.

If, however, the temporal mean is not statistically stationary but evolves in time, it is possible that the averaging interval may not have been extended sufficiently long to include all of the system microstates. On the other hand, it is often the case that the longer-time evolution of the system may consti- tute the very process of interest, and that the short-time averaging is suffi-

Hydrodynamic Equations of the Sea 113

cient to establish the mean and variance of the uninteresting processes. For example, one might wish to study the longer-period variations of oceanic current systems and not be at all interested in the tides, in which event a time series that encompasses a largish number of tidal periods but which is shorter than the characteristic time for change of the large-scale system (which may be months) could be used t o average out the tides, while not doing seri- ous damage to the measurement of low frequency variation. Nevertheless, even though there is a clear spectral gap between the semidiurnaVdiurna1 tides and the motions of interest (see Table 3. l ) , which presumably would allow one to separate out the low frequency signals from the high frequency ones, there are several things potentially troublesome about this procedure.

One is that the tidal motions, through nonlinear effects, can modify the longer term dynamics. Secondly, the tides themselves have frequency components at fortnightly and longer periods and these components will appear directly in the average values. Thirdly, t o thoroughly filter out the tidal components takes a record length of many tens of tidal cycles, during which the slower process itself may well be changing, i.e., the spectral gap is too narrow t o allow the averaging to be done properly. Lastly, for many problems, no spectral gap may exist.

While there is no general solution to this problem except for intelligent consideration of alternatives, a reasonably useful procedure is t o assume the possibility of dividing the motions into slow and fast processes, and to study the consequences for the fluid equations. To d o this, we write the generic fluid quantity as the sum of a mean part, r(x,t0), which is denoted by a cap- ital letter and assumed t o vary on longer time scales, t o , and a fluctuating part, y '(x,?), which is denoted by lower-case primed quantities and charac- terized by higher frequency variation. This latter variation may be either quasi- random or orderly. We will also assume that the temporal and ensemble averages are equivalent in some ergodic sense, so that we may interchange them at will, with the symbolic equivalences being represented by

(3.101) For notational reasons, the bra-ket average is the most convenient. Thus the velocity field, for example, will be decomposed into slowly varying and rapidly varying portions:

u(x,t) = U(x,t,)

+

u ' ( x , t ) , (3.102) where we have made explicit recognition of the low frequency variation of U via the use of to for the slow time. By definition, the mean of u(x,t) over time 2t, is

(3.103)

- u = U = ( u ) ,

while the mean of the fluctuations over the same interval is assumed to vanish:

(3.104)

- u ’ = 0 = (u’).

Similar decompositions and notations can be applied to the other dependent variables in the dynamic and thermodynamic equations.

Our objective now is to derive relationships between the mean and fluctuating quantities on one hand, and the eddy viscosity and diffusivity tensors on the other. During the course of that program we will find that this will impose more precise statistical interpretations on the field variables, as well as improved insight into the meaning of A and K . Additionally, we shall assume that the fluid is incompressible, in which case the continuity equation reduces to Eq. 3.69. Then upon substitution of Eq. 3.102 into the incom- pressibility condition, we obtain

v.

+

^u’)⁼0. (3.105)

Next, we perform our carefully chosen averaging operation on Eq. 3.105, which, being linear, commutes with other linear operators such as differentiation, integration, and addition, so that we obtain from that relationship ( V * ( U

+

^u’))⁼^V.(U)

+

^V*(U’)⁼^{0 .} ^(3.106)

Since (u

’

⁾⁼0, so do each of the partial derivatives making up V (u ’ ), and hence the divergences must vanish separately:

v.u

= 0 = V*(U’). (3.107)

This suggests that both the mean and the fluctuating components of velocity separately behave as incompressible fluids, perhaps, in retrospect, not a sur- prising result.

We now look at the effects of the field decomposition on the momentum equation, which we take to include only the more rigorously derived molecular viscosity, so that A, = Ah = 0; however, we might expect that this param- eterization will reappear during the manipulations. For ease of understand- ing we treat only the x component of that equation in the form of Eq. 3.84:

Hydrodynamic Equations of the See 115

ax ax

Here the horizontal and vertical Coriolis frequencies are, respectively,

f = 20 sin A (3.109)

and

e = 20 cos A

.

^{(3.1 10)}

We next average Eq. 3.108, using the properties expressed by Eqs. 3.103, 3.104, and 3.106, a calculation that yields

au au au au

a t 0 ax aY az

- + U - + + - + W - - f V + e W

acp 1 ap

ax ax

+

^-^-^-

v,v2u

a a a

ax ( u ’ u ’ ) - - aY ( u ’ u ’ ) - - az ( w ’ u ’ )

.

- - -

- (3.111)

The left-hand side of Eq. 3.11 1 is written entirely in terms of the slowly varying “mean” velocity, U, and pressure, P , and their temporal changes during characteristic times, t o . The convective derivative must now be interpreted to be differentiation following this slowly varying motion. The right-hand side, which contains the only surviving terms from the nonlinear advective derivative that involve the fluctuating velocity, is in general non- zero because of the averages of quadratic quantities. These describe momentum fluxes carried by the fluctuations, where in Eq. 3.11 1 the horizontal ( u ’ ) component can transport u ’ , u ’ , and w ’ components of momentum. These terms appear as body forces to the upper-case quantities that tend to reduce the acceleration from other forces, i.e., are frictionlike (unless negative, which happens on occasion), and can be considered to be elements of the divergence of the Reynolds stress tensor, T ~ . The slowly varying flow obeys an equation exactly like the primitive equation (Eq. 3.84) except for the stress divergence on the right-hand side in place of the eddy viscosity. For example, the z-x component of that tensor, which has the form

TLy = - p ( w ’ u ’ )

,

( 3 . 1 1 2 ) represents the average flux of horizontal momentum carried across a sur- face, z = constant, by the fluctuations, in analogy to the arguments leading to Eq. 3.53. The Reynolds stress has a similar matrix representation:

( 3 . 1 1 3 )

I

( u ’ u ’ ) ( u ’ u ’ ) ( U ’ W ’ )

( u ‘ u ’ ) (u‘u‘) ( u ‘ w ’ ) ( w ‘ u ’ ) ( w ’ u ‘ ) ( w ‘ w ’ )

In a statistical sense the elements are to be thought of as the cross-correlation functions of the time-varying velocity components.

The next step in the development is to assume that the stresses are themselves proportional to the macroscopic derivatives of the mean velocity, and that under the influence of those gradients, fluid parcels execute random walks with an effective “mean free path” termed the mixing length, 1 = (lx,ly,l,).

In moving one mixing length, the parcel is assumed to pick up on the average a velocity increment, u’, from the mean flow, which is thereby attenuat- ed somewhat. Such down-gradient fluxes are the simplest assumptions applicable to diffusion processes and work well in kinetic theory of gases;

but in this case of macroscopic fluctuations, the assumptions are more ques- tionable, since there is no method yet known for calculating the coefficients involved. Nevertheless, we will take for the relationships the expressions:

aw

T,, = 2A, - az

,

Hydrodynamic Equations of the Sea

and

117

av a w

az aY

TYZ = T t y = A” -

+

^{A h}^-

.

^{(3.1 14)}

A , and A h are the same quantities as discussed in Section 3.9, and they can be shown to be proportional to the product of the mixing length, the fluctu- ation velocity, and the density, viz:

Ah = P ( I d ’ ) = p < l y u ’ >

.

^(3.1^15a)

and

A , = p ( 1 , ~ ’ )

.

^(3.1^15b)

Thus the diffusion tensor, K = A/p, is seen to be a measure of how rapidly the fluctuating velocity components transport momentum across one mixing length, on the average. The slow-scale vector equation of motion is then

au

1 1

+

U . V U

+

2fl x U

+

^-V P

+

^{V @ =}^-V * A * V U , (3.116)

a

t o P P

which has a form identical to Eq. 3.83, except that the capitalized, averaged variables appear (which change over the longer times, t o ) rather than the in- stantaneous field variables. Thus the eddy viscosity is readily seen to be the result of fluctuations, but more importantly, the momentum equation in the form of Eqs. 3.83 or 3.116 must now be interpreted as one that has been averaged over the high frequency fluctuations, and which is thereby limited to describing the slower evolution of the ocean. What is meant by “high frequency” and “slow evolution” clearly depends on the processes of interest, and thus the numerical values to be used must also depend on them. The efficacy of the algorithm will depend, at the minimum, on the skill of the user, but under the proper circumstances, it appears to work satisfactorily.

For the remainder of this book, we will revert t o the notation of Eq. 3.83 and will not explicitly refer to the averaging of that equation; however, the appearance of eddy coefficients in an equation will carry with it the implica- tion that the equation must be considered to be the same type as Eq. 3.1 16, and that the choice of values for the elements of A and K must be made with the averaging process in mind.

Dalam dokumen Buku Principles of Ocean Physics (JOHN R. APEL, PH.D.) (Halaman 126-134)