Since LESs are inherently stochastic, the natural starting point for subgrid-scale param- eterization is statistical closure theory. We shall use the DIA closure for the barotropic vorticity equations, introduced in Section 3.3.2, to illustrate this approach, following Fred- eriksen and Davies (1997). The two-time covariance equation for the DIA is
∂ ∂tCk(t, t) = −2Re Z t t0 dsηk(t, s)C−k(t, s) + 2Re Z t t0 dsSk(t, s)R−k(t, s) − 2ReD0kCk(t, t) +Fk0(t), (5.21) where ηk(t, s) =−4 X p X q δ(k+p+q)K(k,p,q)K(−p,−q,−k)R−p(t, s)C−q(t, s) (5.22) and Sk(t, s) = 2 X p X q δ(k+p+q)K(k,p,q)K(−k,−p,−q)C−p(t, s)C−q(t, s). (5.23) Furthermore, Fk0(t, t′) =hfk0(t)f−0k(t′)i=Fk0(t)δ(t−s) (5.24) for white noise forcing. Now imagine that a fictitious truncation is performed at some intermediate wavenumber k∗, so that k ≤k∗. We can then split the non-linear terms in Eqs. 5.22 and 5.23 into two parts. One part consists of contributions from wavenumbers pand qsuch that both pandqare less thank∗. This is called the resolved part, denoted by a superscript R. The other part consists of either p and q both greater than k∗, or
§5.3 Closure-based Parameterizations 67
one ofp orqgreater than k∗. This is called the subgrid part, denoted by the superscript
S. Note that the wavenumbersp andq are vectors, so strictly we have to define by what measure we regard one vector to be greater than the other. If we imagine the truncation to be circular, then the circle of radius k∗ defines the boundary between resolved and subgrid wavenumbers. A position vector that lies within this boundary is resolved, and that which is outside is subgrid. With these considerations, we can write 5.22 and 5.23 as
ηk(t, s) =ηkR(t, s) +ηSk(t, s) (5.25)
and
Sk(t, s) =SkR(t, s) +SkS(t, s), (5.26)
Equation 5.21 can then be written as
∂ ∂tCk(t, t) = −2Re Z t t0 dsηRk(t, s)C−k(t, s) + 2Re Z t t0 dsSkR(t, s)R−k(t, s) − 2ReDrkCk(t, t) +Fkr(t). (5.27) Here, Drk=D0k+Ddk (5.28) is the renormalized dissipation, and
Fkr=Fk0+Fkb (5.29) is the renormalized noise covariance. Additionally,
Ddk= 1
Ck(t, t)
Z t
t0
dsηkS(t, s)C−k(t, s) (5.30)
is the subgrid drain eddy dissipation, and
Fkb= 2Re
Z t
t0
dsSkS(t, s)R−k(t, s) (5.31)
is the subgrid eddy backscatter covariance. We can also define the eddy drain viscosity as
νkd= D d
k
k2 (5.32)
and the eddy backscatter viscosity as
νkb =− F b k 2k2C k . (5.33)
The eddy drain and backscatter viscosities can be combined to define a net eddy viscosity
νkn=νkd+νkb. (5.34) The net eddy viscosity parameterizes the net flux between the resolved and subgrid scales. It corresponds to a deterministic parameterization, and as such is closer in spirit to conven- tional ad-hoc eddy viscosities; however, it is calculated self-consistently from the statistics of the flow, and so is an improvement over ad-hoc methodologies.
68 Dynamical Subgrid-scale Parameterizations
The systematic approach to parameterizing the subgrid scales based on the statistics of the flow, as calculated by closure theory, can be traced back to the works of Leith (1971) and Kraichnan (1976). Leith used EDQNM closure theory to calculate a function similar to the net eddy viscosity for atmospheric two dimensional flows. Leith’s dissipation function was used in the general atmospheric circulation models described by Boer et al. (1984) and Laursen and Eliasen (1989) with reported improvements in the simulations when compared to standard diffusion schemes. Kraichnan defined a net eddy viscosity using the test field model (a closure scheme), in both two and three dimensions. He noted that in both cases, the net eddy viscosity has a cusp (a sharp rise) near the truncation scale. Furthermore, in two dimensions, the net eddy viscosity is negative at the large scales, indicating an injection of energy. Kraichnan showed that at the large scales, far away from the truncation scale, the net eddy viscosity is virtually independent of wavenumber. Hence, at the large scales, the net eddy viscosity is simply an enhanced (or reduced) version of the molecular viscosity. Kraichnan’s explanation for this is that at the large scales, the contribution from the subgrid scales is mostly in the form of non-local interactions. Since for any interacting triad k+p+q=0, we can see that ifk≈0 (large scale), then p≈ −q, and since at least one of p or q must be large (that is, a subgrid scale), we conclude that both p and q are large. Hence, the triad interaction involves a large scale mode interacting with two small-scale modes. Now, this is similar to the assumption made in deriving the molecular viscosity, namely, that the motions of interest are large compared to thermal fluctuations. Thus, at the large scales, the net eddy viscosity behaves just like molecular viscosity. However, near the truncation scale, there exist both local and non- local interactions, so the molecular viscosity concept breaks down, and a cusp is seen. In fact, the cusp is an indication that the net transfers are mainly local. We can imagine the following idealized situation. The enstrophy is injected at a single wavenumber, and passed downscale to the next wavenumber only. If there is an abrupt truncation below some wavenumber, then we expect to see a sharp rise in the spectrum only at that wavenumber; other wavenumbers, having passed on the enstrophy to neighboring wavenumbers, will remain unaffected. Hence, to maintain a constant transfer, all we need is a negative transfer at the truncation scale; this is the origin of the cusp seen in the net eddy viscosity. This view is strengthened by the fact that when a smoother filter, for example a Gaussian filter, instead of a sharp cut, is used to effect the truncation, the cusp is not seen (Leslie and Quarini, 1979).
Domaradzki and Rogallo(1990) made the point that it is important to distinguish between energy transfer (between modes) and triad interactions. Although energy transfer between modes is thought to be mostly local in the inertial range, and this is a crucial factor leading to the existence of k−3 (and k−53) power laws in this range, this does not necessarily mean that all the triad interactions that lead to this transfer are purely local. For example, it is possible for one large-scale mode to interact with two small-scale modes of similar size in such a way that the energy (or enstrophy) is exchanged primarily between the two small-scale modes. This is a non-local triad interaction, but the transfer between the two small-scale modes may be quite local. Domaradzki and Rogallo’s results suggest that the transfers occurring in three-dimensional turbulence are predominantly of this type. A similar conclusion was reached by Maltrud and Vallis (1993), who studied non- linear interactions in two-dimensional turbulence. It is important to note that when we refer to local transfers in this thesis, as we shall extensively in Chapters 6, 7, and 8, it is in the sense just discussed and not in the sense that the triad interactions involve wavevectors all of similar size, which is clearly not the case based on the above studies.