5.1. Neptune Effect: Entropic Forcing of Mean Ocean Circulation
This and the following section are drawn from oceans applications based upon this author’s familiarity. Implications for Earth’s atmosphere, other planetary flows and GFD generally will be apparent. First we consider “neptune effect”, so-called after a cartoon [
24]. We will see the strong and largely unsuspected role of entropy organizing Earth’s mean ocean circulation along with a simple opportunity to correct, in part, ocean and climate models.
The theoretical basis for neptune follows from the quasigeostrophic barotropic vorticity equation, one of the fundamentals of GFD, recently discussed [
12] in
Entropy. Further simplifying from [
12], we consider only:
in Cartesian x,y coordinates where
is the vertical component of relativity vorticity and
ψ is the corresponding streamfunction for horizontal, nondivergent (in quasigeostrophic approximation) velocity,
u. ∂(,)/∂(,) denotes the Jacobian determinant. The fluid, bounded above and below by rigid level planes is subject to uniform background rotation about vertical axis
.
h(
x,
y) appears in (8) when we allow one of the bounding planes to include small (with respect to separation between planes) deformations. Thus
h(
x,
y) may represent bottom topography underlying an atmosphere or ocean in this approximation. At this point we put the right side of (8) to zero,
i.e., considering only unforced, non-dissipative advection of potential vorticity,
q =
ζ +
h.
There are two reasons to begin so simply as (8). First, we are isolating only mechanisms that allow rapid adjustments by advecting q.
Second, and possibly of special interest to readers of
Entropy, we connect with two of the seminal papers of Onsager [
25,
26]. [
25] considered (8) without
h,
i.e. supposing
h=0, while treating the vorticity field as a collection of pointwise vortices each carrying integrated vorticity, termed circulation, Г
i ≡ ∫
dAζi where Г
i is the circulation due to vorticity,
ζi, of the
ith point vortex and
dA is an elemental area. (The problem is posed in two dimensions.) The collection of point vortices conserves interaction energy:
where
rij is the separation between
ith and
jth vortex. Depending on configurations of the vortices,
E can take any value from - ∞ to ∞. But then the available phase space (here configuration space) is a function of
E, taking a maximum at some
E*. For
E >
E*, greater
E would decrease available phases space, a circumstance of “negative temperature”. The result is that vortices of like sign tend to cluster, creating larger compound vortices (today we say “coherent structures”) while weaker vortices are more free to roam nearly randomly. In this way Onsager foresaw much of the literature of “two-dimensional turbulence” that would unfold over the subsequent half century.
[
26] considered a different question: the orientation of rod-like particles in colloidal suspension. Entropy in this case depends upon the freedom of each rod to be nearly randomly located and nearly randomly oriented. If the density of rods is increased, there comes a point where restricting the freedom of orientation by forming patches of like-oriented rods (“nematic ordering”) makes available greater volume for more nearly random location. In this way Onsager foresaw much of the literature of “excluded volume effects” that play a role today understanding entropic forces in colloidal chemistry, microbiology and nanomechanics.
Onsager did not happen to include the role of bottom topography,
h in (8), when he considered vortices. But we can imagine only a footnote in [
25] if Onsager appended a collection of bound vortices (assigned to fixed locations) with circulations Ω
k. Then there is interaction energy Г
iΩ
klog
rij as well as Г
iГ
jlog
rij (energy due to products ΩΩ is a constant for fixed geometry of bound charges). If free vortices tend to locate near bound vortices of like sign, increased energy stored in ГΩ permits reduced energy in ГГ, allowing greater mutual freedom of location among the free vortices. Analogously colloidal particles in containers with boundaries that include concave and convex variations will tend to cluster near (or avoid) fixed locations of preferred concavity, driven by entropic forces arising from excluded volume effects. In oceans, currents will tend to locate their vorticity in specific regions given by bottom geometry, allowing greater numbers of “free” vortices to roam more randomly. Entropic ordering, foreseen by Onsager and observed at microscales, thus appears in oceans at megascales.
[
8] considered statistical equilibrium for circumstances more complicated than (8) by including motion in two layers [only one layer exists in (8)] and including variation of Coriolis (vertical component of planetary rotation) with latitude. Sufficiently for present purpose, a concise treatment [
27] limited to (8) is summarized below.
Whereas [
25] adopted the idealization of assuming point vortices, instead we expand:
on a set of continuous eigenfunctions defined by
. Equation (8) conserves energy,
E, and potential enstrophy,
Q:
Maximizing (1):
where Lagrange multipliers,
αj, impose constraints 〈
E〉 =
E0, 〈
Q〉 =
Q0, and 〈1〉 = 1, with 〈〉 denoting expectation. Consequently log
p + 1 +
α1E + α2Q +α3= 0 or:
where:
or:
with
α1,
α2, and Ω given by constraints on 〈
E〉, 〈
Q〉 and 〈1〉.
The probability density (12) is Gaussian due to quadratic constraints (10). Further constraints can be applied leading to departures from Gaussianity,
viz., [
28,
29,
30]. However, what is especially important here is that the Gaussian (12) is centered not about zero but about the mean flow given by (14). If one imagines initializing an ideal “ocean” defined by (8) with random currents with zero mean, the currents will spontaneously organize
. This is exactly the opposite of what is assumed in atmosphere - ocean - climate models where the action of SGS eddies is presumed to erode any mean flows toward zero. We should pause at this, realizing that large models which are guiding public policy are based on assumptions such as SGS “friction” that seem to be quite mistaken.
The result (14) is not applicable in real oceans for at least two reasons. First, (14) is an equilibrium (ME) result, supposing an “ocean” with no internal dissipation, isolated from all external forcing. Clearly that is far from real forced, dissipative oceans. Applications in real oceans or atmospheres must follow from disequilibrium statistical mechanics, not ME. Second, even apart from forcing and dissipation, actual dynamics of oceans or atmospheres are vastly more complex than (8).
What to do? The answer is
not to revert to traditional “friction” for want of a better idea. What we learned in (8) - (14) is that oceans can generate entropy if their currents become closer to
. That potential to increase entropy should appear as an entropic forcing which drives ocean currents and which was missed in the classical mechanical basis for GFD. The challenge is to construct a serviceable approximation. Recognizing uncertainty about calculations far from ME, and also recognizing that effective parameterizations for models must be as simple as possible, the following was proposed [
27].
Streamfunction
ψ, which may be interpreted as either velocity or transport streamfunction in (8) where quasigeostrophy is ambiguous, is taken in the sense of transport:
where
z = − D defines the sea bottom and
z is unit vector in vertical. In (14),
α1/
α2 ≡ 1/
L2 defines a lengthscale parameter
L. For the idealization (8), with given set of
φn and given
E and
Q,
L can be evaluated exactly. In oceans this is not possible and
L is adjusted to obtain pleasing model outcomes. However, the values of
L so found, ranging from O(10 km) in open ocean subtropics to just a few km at high latitudes or in shallow and semi-enclosed seas plausibly reflect a lengthscale below which eddy vorticity fluctuations rapidly diminish. We may use this information to greatly simplify (14) given that ocean models typically compute on grids very much larger than
L. Then
α1/
α2 (= 1/
L2) very much dominates ∇
2 and we simplify (14) by omitting ∇
2, avoiding the need to invert (
α1/
α2 − ∇
2) and obtaining simply
. On this basis a neptune transport streamfunction is defined as Ѱ* = −
fL2D, where the negative sign is because
h is traditionally expressed as a small elevation above the mean bottom, scaled by the mean depth and multiplied by Coriolis parameter,
f. From Ѱ* and (15) we obtain neptune velocity:
After L is assigned, u* is given from ocean basin geometry (D). u* is independent of depth, z, and time, t. Clearly u* is not itself a very good descriptor of oceanic u(x,y,z,t). However, the difference field, u* − u, as a component of Y* − Y, cf. (7), points to a higher entropy configuration for u which is relatively accessible by rapid (quasigeostrophic vorticity advection) processes. The accessiblity of that higher entropy u is the source of entropic forcing which should appear in momentum equations but which is missed in the classical mechanical basis for GFD.
It remains to represent
K. Closure theory such as [
12] might be applied. Fine resolution numerical experiments can be performed to obtain empirical
K. In modeling practice the result must also be made rather simple and computationally inexpensive.The neptune compromise, to date, has favored
K=
A∇
2 with
A a constant coefficient. Then a modeled horizontal eddy viscosity,
Ah∇
2u, is simply replaced by
Ah∇
2(
u–u*).
In the vertical, usual viscous terms may be expressed
∂z(
Av∂zu), where
Av is a vertical viscosity typically presumed to depend upon
z. Because
u* is independent of
z (but see
Section 6), the form of vertical momentum mixing is not changed. However, the magnitude of
Av changes enormously. Usual modeling supposes that momentum is “mixed” similarly with fluid parcels, hence with a coefficient comparable to (usually rather larger than) water property mixing coefficients. Neptune recognizes vastly more effective ways of rearranging momentum by potential vorticity advection, with
Av typically orders of magnitude greater than as usually assumed.
Preceding paragraphs have given a rationale for, and brief summary of, neptune as practiced. It should be clear that many of the steps should be refined, possibly in conjunction with closure theory such as [
12] (
Section 3). Such research remains to be done and is beyond present scope. Here we only summarize recent results from neptune then return to open research questions in
Section 6.
Modeling experiments with neptune began with [
31,
32] and are ongoing. A review [
33] has discussed much of the earlier work. We recall only one item from earlier work because of its relevance to discussion in
Sect. 4, comparing MEP with the present entropy gradient estimations. [
34] examined MEP after [
22] for an Arctic Ocean model similar to a case studied by [
35] using entropy gradient forcing. Pleasingly, the two results were rather similar, suggesting that differences between MEP and entropy gradient estimations are not crucial. Of course this single examination was far from exhaustive.
Below we consider work since [
33].
A major project (Arctic Ocean Models Intercomparison Project (AOMIP), [
36]) involved efforts of 15 Arctic Ocean modeling groups in nine countries, examining differences among models, and differences between model results and observations, under conditions of similar setup and forcing. Within AOMIP, [
37] compared temperature, salinity and velocity fields among nine of the models. Three of the nine models included neptune while the remaining six employed more traditional frictional representations. Comparing velocity fields among models is difficult because of the complicated vector fields varying in three dimensions and time. A descriptive measure termed topostrophy,
τ ≡f x
u·∇
D, was introduced, where
f is the vertical Coriolis vector,
u is model velocity, and ∇
D is the gradient of total depth,
D. In this way the complicated vector field
u and complicated basin geometry
D were combined in a single scalar variable which could be averaged over various regions. Results were startling. Whereas all measures show differences among models,
τ seemed to separate the models into disjoint classes as illustrated in
Figure 1. Here
τ is averaged over the Eurasian Basin (a portion of the Arctic Ocean) and normalized:
where brackets {}denote averaging over a region.
Symbols in
Figure 1 designate specific models, whose identities are not important here. What is important is that all of the traditional models produce highly variable timeseries for
with values bounded -.4<
<+.4 and most
amplitudes smaller than about 0.1 with frequent reversals of sign. Contrariwise the three neptune models exhibit relatively persistent
in the range +.4<
<+.7. Averages over other regions within the Arctic or over the entire Arctic Ocean showed very similar results. Although
Figure 1 only shows timeseries of an averaged scalar, examination of the model
u maps (not shown here) reveals stunningly different perceptions of how Arctic Ocean currents may flow. Each of the neptune models show persistent cyclonic (counter-clockwise) “rim currents” around the several Arctic basins whereas traditional frictional models tend to be ambiguous, showing broader gyre circulations of variable cyclonic and anticyclonic sense.
Figure 1.
Normalized topostrophy averaged over the volume of the Arctic Eurasian Basin is plotted for nine AOMIP models [
37].
Figure 1.
Normalized topostrophy averaged over the volume of the Arctic Eurasian Basin is plotted for nine AOMIP models [
37].
The principle result from
Figure 1 is that the possible role of entropic forcing, however imperfectly realized in the neptune parameterization, can be quite strong in comparison with the classical forces represented in traditional modeling. At the date of publication (2007), it was not known which trace in
Figure 1 might be closer to reality. Subsequently, from a world database of more than 17,000 long term current meter records spanning over 83,000 current meter-months, climatological
was mapped globally as function of latitude and depth [
38]. These observations clearly put regionally-averaged Arctic
>+.4. The strong suggestion, as seen also in many cases cited in [
33], is that traditional ocean modeling, based upon classical GFD, is systematically deficient.
There is one further note from the current meter observations that may be of special interest to readers of Entropy. Globally
is positive with overall mean value near +.3 (based on available data). It has long been understood in classical GFD that mean flows, under the influence of background rotation, tend to follow contours of bottom topography. Hence f x u tends to align with ∇D. But either sign for f x u·∇D is allowed. Why does observed
so clearly favour positive sign? As we’ve seen, entropic forcing introduces the Second Law (2), supplying classical GFD with an “Arrow of Time”, thereby selecting the sign of
.
Three more recent publications should be mentioned. [
39] examined topostrophy among four global ocean models: two with relatively coarse grid spacing, two with finer grid spacing. None of the four employed neptune. The two models with finer grids tended to produce somewhat more positive topostrophy.
[
40] explored introducing neptune into a global model with sufficiently fine grid that the model spontaneously generated eddies (albeit poorly resolved). In such models it is usual to replace Laplacian friction with a biharmonic form (
i.e., ∇
2 with ∇
4). Correspondingly, neptune was represented as
A∇
4(
u−
u*). The interesting question is: if realized eddies are already generating entropy, what role is left for an entropy generation parameterization? Comparing model runs with and without
u*, [
40] obtained modest improvements with
u* present, suggesting that explicit eddy generation was not fully represented.
Finally, returning to the Arctic, the model “OPA-LIM” [
41,
42], not previously included in AOMIP studies, was examined [
43]. Consistently with prior AOMIP results,
cf.
Figure 1, markedly higher
was found [
43] when OPA-LIM was modified for neptune. This study also allowed detailed examination of entropically forced cyclonic “rim currents”.
5.2 Differential Mixing of Heat and Salt in the Ocean
We turn to a very different example on very different scales in order to see how robust are some of the entropic insights. The example is chosen because it illustrates another circumstance where simpler intuition has led to mistaken ideas, and entropic calculus reveals surprising understandings with practical, observable consequences.
We contemplate mixing heat and salt by small scale turbulence in the gravitationally stably stratified ocean. Larger scale motions are regarded as internal gravity waves. Occasional superpositions of waves yield local regions of instability (“breaking”) where turbulent mixing occurs. Energy is stored both as kinetic energy and as gravitational potential energy, with spectral densities
U(
k) and
B(
k) respectively. Here we will treat
k merely as magnitude wavenumber. In fact, motions on larger scales are quite anisotropic, and one should at least distinguish vertical and horizontal wavenumbers. At smaller scales, even during more intense turbulent events, anisotropy tends to persist. This is a complication to which the reader should be aware, but which will not overly concern us for the present level of discussion. Potential energy arises because heat and salt affect the density of sea water due to volumetric coefficients of thermal expansion and haline contraction. It is convenient to define buoyancy,
b(
x,
t) =
ρ0(
z,
t)−
ρ(
x,
t), where
ρ is density and
ρ0 is regionally horizontally averaged
ρ. Gravitational potential energy occurs due to fluctuations in
b such that specific potential energy (per unit mass) is
B = ½
gZb2/
ρ02 where
g is the acceleration of gravity and
Z= (
∂zlog
ρ0)
-1 is a height scale due to stable background stratification in
ρ0. Specific kinetic energy is
U = ½
u·
u. Spectral energy balance equations for
B and
U are:
and:
where
χ(
k) is transfer (by nonlinear effects) of
B from wavenumbers less than
k to wavenumbers greater than
k,
ε(
k) is the corresponding transfer of
U,
η(
k) is the dissipation of
B due to thermal and haline conduction, and
ξ(
k) is the dissipation of
U due to viscosity. Importantly,
F is an exchange of energy between
B and
U by vertical buoyancy flux,
F = gwb/
ρ0, due to correlations of vertical velocity,
w, with
b.
Traditional understanding has characterized large scale motion as interaction among waves until, at some smaller scale, waves “break” and turbulence ensues. At the scales of wavelike motion, F should vanish on average since individual waves carry no mean F and the waves are believed to be nearly in random phase superposition. At the smaller scales of turbulence, we might expect a downward mixing of lighter (more buoyant) water above with denser water below, i.e., F<0. Such downward mixing is necessary to compensate the gradual overall upwelling of deep water that has been supplied by sinking (mainly at high latitudes).
Theoretical approaches are possible. On large scales and assuming waves are sufficiently weak, closure methods described in
Section 3 can be applied to waves - turbulence mix. In small amplitude limit, [
44] showed that such closures approach the resonant interaction approximation (RIA) seen,
e.g., in [
45,
46]. For RIA, applied to statistically homogenous fields, systematic energy transfers occurs only on triads for which wavevectors
k +
p +
q = 0 and also the intrinsic frequencies of the three wave components must satisfy resonance
ωk +
ωp +
ωq = 0. At larger amplitude, the condition of frequency resonance is somewhat relaxed as near-resonant waves support part of the energy transfer. Finally, in the large amplitude limit, frequency resonance becomes entirely insignificant and all waves participate in energy transfer. Importantly from an entropic view, these closures ranging from RIA to full turbulence strictly satisfy the Second Law (2).
RIA has been explored by [
47,
48] and others to attempt to account for the observed spectral distribution of oceanic internal wave energy. On the scales for which RIA would be valid,
F vanishes. In fact the oceanic scales for which RIA is applicable remain an unresolved question,
viz., [
49,
50]. Closure theories capable of representing strongly interacting waves and turbulence (i.e., beyond RIA) have been examined by [
51,
52,
53,
54].
Full examination by closure theory has been to date too daunting, and the above cited works have made partial accounts of the spectral distributions of kinetic or potential or wave energy. Vertical buoyancy flux
F was considered [
53] but only under the very restrictive and unrealistic constraint of motion confined to a vertical plane. Let us here see how the concept of entropy gradient forcing,
C·∂YH in (6), can help us foresee the more complete picture despite specific uncertainties that will remain for closure theory and/or numerical experimentation.
In equations (18) for ∂
t(
B,
U) we seek to represent terms arising from
C∂H/∂(
B,U) where
∂H is change of total entropy with respect to changes
∂(
B,U) in
B and
U while
C is an operator projecting these entropy tendencies onto ∂
t(
B,U). Here, although scales of motion are very much closer to scales of molecular chaos (compared with
Section 5.1), we can still proceed without taking direct account of molecular contributions to
H [included parametrically as
η and
ξ in (18)].
Omitting external forcing and internal dissipation, dynamics conserve the total energy,
B+U. ME thus favours energy equipartition. This is far from reality in which large scale forcing and small scale dissipation lead to spectra steeply decreasing with increasing wavenumber
k. The result is a strongly entropically driven forward transfer of both
B and
U from small
k toward large
k. Traditional scaling arguments from turbulence are not so directly applicable due to additional parameters that have entered on account of gravitationally stable background stratification,
cf. [
55,
56,
57,
58,
59].
At small scales (large
k), one might imagine approaching classical (unstratified) turbulence. For the moment we consider
B and
U. Later we return to
F.
B and
U behave similarly on scales for which turbulent advection dominates both. Writing for
U, nonlinear transfer:
where
U* is constant over modes. Operator
K is sometimes hypothesized as a sort of diffusion, K≈
∂kDU∂k, with
k-dependent coefficient
DU. With (19), this suggests
ε ≈ −
DU∂kU. Continuing for this moment with classical turbulence ideas, if we hypothesize that
DU depends only upon “local” scale
k and
ε(
k), we must take
DU ≈
ε1/3k8/3 ≡
τ-1k-2/3, where an “eddy timescale” is
Then:
a classical result since Kolmogorov [
60], albeit argued differently by different authors. Considerations for
B are similar to those for
U on scales for which advection dominates, hence B ≈ χ
2/3k-5/3 on scales for which
η and
ξ in (18) are not important while we continue, for this moment, to set aside
F. Further considerations arise because the ratio of viscosity,
v, to thermal conductivity,
κ, is near
v∕κ ≈ 7 while the ratio of viscosity to haline conductivity,
γ, is near
v∕γ ≈ 700 for seawater. Thus, fluctuations in temperature or salinity persist to smaller scales (larger
k) where velocity fluctutations are suppressed by viscosity. On these scales,
k >
kK where
kK =(
ε∕v3)
1/4, Batchelor [
61] argued that the eddy timescale should be given by a characteristic straining rate (
ε∕v)
1/2 due to velocity fluctuations near
k ≈
kK, yielding B ≈
χ(
ε∕v)
1/2k-1. Over the broader range of scales including
k <
kK, this yields:
neglecting so far the role of stable stratification.
A reader may feel alarmed in the previous paragraph for the many uses of “≈” and other “loose” arguments. While this is due in part to necessary brevity in the present paper (more complete discussion having filled literature for decades), it is appropriate to recognize that further careful work remains. Only for present purpose the preceding “thumbnail” may capture some of classical turbulence theory sufficiently to let us proceed. Notably for readers of Entropy, we see that these “cascades” of B and U are driven by entropy generation, ultimately expressed at the level of molecular chaos.
At some sufficiently large scale (small
k), it may be that RIA describes wave-wave energy transfers,
viz. [
47,
48]. While neither the RIA limit nor the turbulent limit may be fully realized in the ocean, a greater challenge yet is to bridge the intermediate scales with mixed wavelike and turbulent-like properties. In closure theory a crucial quantity is termed “triple correlation timescale”,
θk,p,q, characterizing the time over which three modes satisfying
k+
p+
q=0 can remain phase-correlated to enable systematic energy transfers. A bridge between wave interactions and turbulence was seen [
44] where:
limits on
πδ(
ωk +
ωp +
ωq) in a weak wave limit,
μ ∕ ω → 0, and upon (
μk +
μp +
μq)
-1 in a turbulence limit,
ω ∕ μ → 0, where
μ is a nonlinear deformation rate obtained from the closure theory. It is this full evaluation from closure theory that has, to date, remained daunting.
We can push a little ahead with some simplifying, albeit uncertain, suppositions. If we suppose the energy transfers
ε and
χ are only weakly divergent in
k on scales for which dissipation rates
η and
μ are small,
i.e., supposing that buoyancy flux
F does not dominate, then we may associate
μ(k)
≈ τ-1(k) with as (20). At intermediate scales, between weak waves and strong turbulence, we suppose precise resonances
ωk +
ωp +
ωq ≅ 0 are not crucial and estimate for typical triples of waves (
ωk +
ωp +
ωq)
2 is “roughly”
N2 where
N is termed “buoyancy frequency”,
N2 = −
g∂zlog
ρ0. Near a scale
k, we estimate (23) simply as
θ ≈ τ-2∕(
τ-2 +
N2). From closure theories we know energy transfers are proportional to
θ. Thus, if we appeal to the simple heuristic
ε ≈ −DU∙∂kU, then take
DU∙ ≈ θτ-2k2 which, with the expression for
θ, yields:
and similarly:
apart from the direct roles of dissipation at high
k and the effects of near-resonances for wavelike interactions at low
k. Spectra near (24) are observed in oceans [
62] and in the middle atmosphere [
63].
For clarity of illustration,
Figure 2a shows
k2U,
k2BT and
k2BS, where we assume (for reasons to be explained shortly) that total buoyancy,
B, is contributed in a representative oceanic example 60% from temperature fluctuations yielding
BT and 40% from salinity (concentration of dissolved ions) fluctuations yielding
BS.
In obtaining (24), we’ve not especially depended upon entropic forcing ideas. We see “forward” (i.e., from large scales or low k to small scales or high k) energy transfers,
ε > 0 and
χ > 0, driven by the relative excess of energy at large scales compared with small scales (i.e., far from ME). The different dependences upon k in the terms in (24) reflect different processes that affect the efficiencies of transfers. Significantly, we know [
15] that closure theories which lead to (24) are proven to satisfy (2). Nonetheless, we could have inferred these spectra without considering entropy.
The subject turns unexpected, indeed startling, when we inquire about
F. Recall that traditional thinking supposes
F ≈ 0 on larger, more wavelike scales with
F < 0 (
i.e., downgradient buoyancy mixing) on smaller, more turbulent scales. Turbulence is usually thought to occur on scales for which density structures are occasionally overturned, observed to occur near a scale
kb = (
N3∕
ε)
1/2 where spectra (24) turn from
k-3 to
k-5/3. Efforts to observe downgradient
F have been frustrating. Laboratory experiments [
64,
65,
66] and numerical simulations [
67,
68,
69] often suggest just the opposite,
F > 0, termed “persistent countergradient fluxes”. Why?
A tendency toward energy equipartition (here ME) occurs not only across scales of
k but also among the available modes of motion at any
k. Linear modes from (4), assuming incompressibility, ∇∙
u = 0, include two inertio-gravity waves (oppositely propagating) and a geostrophic mode (sometimes called “vortical”) mode. When these three are equally excited, then
U = 2
B at each
k [
70]. A reader might also intuit this by realizing that only two components of
u are independent, given ∇∙
u = 0, while a third variable is
b, so that energy is shared among two velocity components and one buoyancy component at each
k.
Figure 2.
(a) Spectra k2BT (solid line), k2BS (dashed line) and k2U (dash-dot) are shown. (b) Spectra k2FT (solid line) and k2FS (dashed line) are shown where FT and FS are contributions to total buoyancy flux F=w’b’ due to temperature, T, and salinity, S. Scalings are assumed such that buoyancy wavenumber kb=1 while Kolmogorov kK and diffusive cutoffs kT and kS for T and S are representative of corresponding typical oceanic values.
Figure 2.
(a) Spectra k2BT (solid line), k2BS (dashed line) and k2U (dash-dot) are shown. (b) Spectra k2FT (solid line) and k2FS (dashed line) are shown where FT and FS are contributions to total buoyancy flux F=w’b’ due to temperature, T, and salinity, S. Scalings are assumed such that buoyancy wavenumber kb=1 while Kolmogorov kK and diffusive cutoffs kT and kS for T and S are representative of corresponding typical oceanic values.
There develops a competition between entropy generating processes. On one hand, excess energy (both
U and
B) at low
k drive positive transfers
ε and
χ. But the transfers are not equally efficient. Even in the absence of gravitationally stable stratification, it is well known from closure theories (
viz., [
71]) that transfer of tracer variance, here
b2, is more efficient than transfer of velocity variance,
u∙
u. The result is to retain somewhat more kinetic energy,
U, in the sense
U > 2
B at low
k while accumulating somewhat more potential energy,
B, in the sense 2
B >
U at high
k. Then
F arises as the entropic forcing,
C∙∂YH, at each local
k driving toward
U = 2
B. At lower
k, |
F| will be impeded by wave propagation tendencies.
Quantifying details of
F will require closure theoretic investigation beyond present scope, including taking account of anisotropy of
B and
U. With plausible assumptions, [
72] obtained
F as seen in
Figure 2b. Again for clarity the figure shows
k2FT and
k2FS where
FT and
FS are contributions from
T and
S in the representative oceanic case shown in
Figure 2a. The choice to weight by
k2 allows one to better see the countergradient fluxes
F < 0.
Details of this
F are uncertain. We cannot say,
e.g., at precisely what
k the sign of
F changes from
F < 0 at lower
k to
F > 0 at higher
k, and we do not know how effectively wave propagation suppresses |
F| at yet lower
k [
73]. What is stunning though is how strongly this theoretical result contradicts traditional thinking. We anticipate
F < 0 over scales for which it was supposed
F ≈ 0. We anticipate
F > 0 over scales for which it was supposed F < 0. Thus “turbulence”, over
k >
kb is not expending kinetic energy to mix buoyancy downward but rather the turbulence is being excited by release of potential energy as buoyancy “re-stratifies” (upwards flux). Is all of this very strange? No.
F is only taking the sign of positive entropy production,
i.e., following the Arrow of Time.
With such clear contrast between present theory and traditional thinking, can the difference be tested? It’s not easy. Although laboratory experiments and numerical simulations have shown persistent countergradient fluxes, F>0, there are always questions how any experiment (laboratory or numerical) is set up. Interestingly, the oceans offer another test, possibly with greater consequences.
As mentioned at
Figure 2a, we illustrate for the case of buoyancy in seawater which is controlled by both temperature,
T, and salinity,
S. However the molecular coefficient,
κT, for thermal conduction is about 100x greater than the coefficient,
κS, for ionic conduction. There are regions of the ocean which may be gravitationally stably stratified with respect to one tracer while unstably stratified with respect to the other tracer while the total density remains stable,
∂zρ0 < 0. This leads to many interesting phenomena, generally termed “double diffusive”. Throughout much of the oceans, stratification is stable with respect to both
T and
S,
i.e.,
∂zT > 0 and
∂zS < 0. Then the usual practice supposes that turbulent mixing is the same for
T and
S in the sense that vertical fluxes
w’T’ and
w’S’ are represented by the same eddy diffusion coefficients,
AT and
AS.
I.e.,
γ ≡ AS/
AT = 1.
If traditional views about turbulent mixing in stably stratified flows were correct, we would expect small scale turbulence to support downgradient fluxes. The lesser molecular conduction coefficient,
κS, would allow a somewhat wider range of downgradient flux,
w’S’, hence
γ < 1. On the other hand, if the arguments for
F from entropic forcing are correct, then we expect small scale turbulence to be characterized by countergradient (upgradient)
w’T’ and
w’S’ which will be partially offsetting downgradient fluxes from larger scales. In this case a somewhat wider range for
w’S’ permits somewhat greater offsetting flux for overall
γ < 1. Oceanic observations [
74], laboratory studies [
75], and numerical experiments [
69,
76] have found
γ < 1 in gravitationally stable environments.