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Positive linear stratified energy and diffusion

Statement

Let $I \subseteq R$ be a nonempty open interval, assume the Axiom of Countable Choice ${A C}_{ω}$ , and use the periodic domain $T^{3} = R^{3} / (2 π Z)^{3}$ with Lebesgue measure $λ_{3}$ on $Q_{3} = [0, 2 π]^{3}$ , where $d x = d λ_{3} (x)$ and the time-space integral is iterated with the spatial integral first. Let $f \in R$ , $N > 0$ , and $ν, κ \geq 0$ be constants. Suppose $u, b$ are continuously differentiable in time and space with continuous second spatial derivatives, and $π$ has continuous first spatial derivatives; all fields are $2 π$ -periodic and satisfy

$u_{t} + f e_{3} \times u + \nabla π = b e_{3} + ν Δ u, \nabla \cdot u = 0, b_{t} + N^{2} u_{3} = κ Δ b .$

Define the positive energy per reference density

$E (t) : = \frac{1}{2} \int_{Q_{3}} (∣ u (t, x) ∣^{2} + \frac{b (t, x)^{2}}{N^{2}}) d x .$

Then for every $t_{0} < t_{1}$ in $I$ ,

$E (t_{1}) - E (t_{0}) = - \int_{t_{0}}^{t_{1}} \int_{Q_{3}} (ν ∣ \nabla u ∣^{2} + \frac{κ}{N^{2}} ∣ \nabla b ∣^{2}) d x d t .$

Thus the inviscid model conserves $E$ , and the stated nonnegative diffusion coefficients make it nonincreasing. Here $N > 0$ fixes the positive buoyancy-frequency convention; the weight $1 / N^{2}$ is positive for every nonzero real $N$ , while the displayed functional is undefined at $N = 0$ . This is an exact identity conditional on the displayed smooth linear Boussinesq equations; it is not a nonlinear or compressible stability theorem. A discrete analogue requires a skew Coriolis operator and compatible pressure/divergence pairing in the actual discretization; no claim is made for arbitrary schemes.

Statement references

Equation family and pointwise regularity: Rotating stratified and shallow-water equation families.

Continuous functions on closed rectangles are Riemann integrable: Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable.

Under ${A C}_{ω}$ , bounded Borel Riemann integrands on finite-dimensional boxes have the same Lebesgue and Riemann integrals: Borel Darboux integrands in finite dimension.

Riemann integration over products of rectangles and equality with iterated integrals: Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections.

Newton–Leibniz for a continuous endpoint function with an integrable interior derivative: Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative.

Under ${A C}_{ω}$ , bounded Riemann integrable functions on closed intervals are Lebesgue integrable with the same integral: A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral.

Linearity of the multidimensional Riemann integral: Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ .

Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, §1.6.2–1.6.3, pp. 15–17: full lecture notes, PDF.

Facts & Assumptions

Given: $I \subseteq R$ is nonempty and open; the Axiom of Countable Choice ${A C}_{ω}$ ; $u, b, π$ are classical and $2 π$ -periodic in each spatial coordinate; $f \in R$ , $N > 0$ , and $ν, κ \geq 0$ ; the displayed equations hold pointwise.

[F1]

Rotating stratified and shallow-water equation families: the stratified branch on $T^{3}$ is $u_{t} + f e_{3} \times u + \nabla π = b e_{3} + ν Δ u$ , $\nabla \cdot u = 0$ , and $b_{t} + N^{2} u_{3} = κ Δ b$ , with constant $f \in R$ , $N > 0$ , and $ν, κ \geq 0$ .

[F2]

Rotating stratified and shallow-water equation families: for a scalar field $q$ , $\nabla q = (\partial_{x_{1}} q, \partial_{x_{2}} q, \partial_{x_{3}} q)$ ; for a vector field $w$ , $\nabla \cdot w = \sum_{j} \partial_{x_{j}} w_{j}$ , $Δ w$ is componentwise, $∣ w ∣^{2} = \sum_{j} w_{j}^{2}$ , and $∣ \nabla w ∣^{2} = \sum_{i, j} (\partial_{x_{j}} w_{i})^{2}$ ; $e_{3} \times w = (- w_{2}, w_{1}, 0)$ .

[A1]

The fields satisfy these equations with the regularity in the statement. The energy density, its time derivative, and the dissipation integrand below are continuous on each compact time interval times $Q_{3}$ . At each fixed time, the pressure and every spatial flux derivative below are continuous on $Q_{3}$ ; no continuity in time of the pressure itself is assumed.

[A2]

Countable Choice ${A C}_{ω}$ means that every $N$ -indexed family of nonempty sets has a choice function; it is used for the cited Lebesgue-measure and Riemann-to-Lebesgue comparison results (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

Every continuous real function on a closed nondegenerate rectangle in $R^{m}$ , $m \geq 1$ , is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable).

[L2]

Assuming ${A C}_{ω}$ , every bounded Borel Riemann-integrable real function on a closed nondegenerate box is Lebesgue integrable and its Lebesgue and Riemann integrals agree (Borel Darboux integrands in finite dimension).

[L3]

For a Riemann-integrable function on a product of nondegenerate closed rectangles, if every section in a chosen order is Riemann integrable, the ordinary iterated Riemann integral in that order exists and equals the product Riemann integral; integrability on the product alone does not guarantee that every section is integrable (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[L4]

If $G$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $G^{'}$ has a Riemann-integrable extension $f$ to $[a, b]$ , then $\int_{a}^{b} f = G (b) - G (a)$ (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative).

[L5]

Assuming ${A C}_{ω}$ , a bounded Riemann-integrable real function on a closed interval is Lebesgue integrable and has the same Lebesgue and Riemann integral (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral).

[L6]

The multidimensional Riemann integral is linear on integrable functions over a nondegenerate rectangle (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

Proof

1.1givenF1F2A1algebra

Dot the momentum equation with $u$ . Since $u \cdot (e_{3} \times u) = 0$ pointwise and $u \cdot \nabla π = \nabla \cdot (π u)$ by the component definitions in [F2] and $\nabla \cdot u = 0$ , this gives $\partial_{t} (∣ u ∣^{2} / 2) + \nabla \cdot (π u - ν \sum_{i = 1}^{3} u_{i} \nabla u_{i}) = b u_{3} - ν ∣ \nabla u ∣^{2}$ . Here $u \cdot Δ u = \nabla \cdot (\sum_{i} u_{i} \nabla u_{i}) - ∣ \nabla u ∣^{2}$ follows by the product rule, component by component.

1.2A1A2L1L2L3L5

Fix $t_{0} < t_{1}$ and set $K = [t_{0}, t_{1}] \times Q_{3}$ . For each rectangle $A$ , write $R_{A} (F)$ for the Riemann integral of a Riemann-integrable $F$ over $A$ . The energy density, its time derivative, and the dissipation integrand are continuous on $K$ and on its closed coordinate subrectangles; hence they are bounded and Riemann integrable by [L1], and Borel because the inverse image of an open set is relatively open. Under [A2], [L2] identifies their Riemann integrals with their Lebesgue integrals on $K$ and on each spatial slice. For any such continuous $F (t, x)$ , [L3] gives a Riemann-integrable section integral $g_{F} (t) = R_{Q_{3}} (F (t, \cdot))$ and $R_{K} (F) = R_{[t_{0}, t_{1}]} (g_{F})$ ; [L2] identifies each spatial section integral with its Lebesgue integral, and [L5] identifies the outer interval integral of $g_{F}$ with its Lebesgue integral. Thus the nested Lebesgue dissipation integral in the statement exists and equals its product Riemann integral on $K$ .

2.1givenF1F2A1step 1.1algebra

Multiply the buoyancy equation by $b / N^{2}$ and use $b Δ b = \nabla \cdot (b \nabla b) - ∣ \nabla b ∣^{2}$ to get $\partial_{t} (b^{2} / (2 N^{2})) + \nabla \cdot (- (κ / N^{2}) b \nabla b) = - b u_{3} - (κ / N^{2}) ∣ \nabla b ∣^{2}$ . Adding this to step 1.1 cancels the two $b u_{3}$ terms and yields a local energy balance with flux $π u - ν \sum_{i} u_{i} \nabla u_{i} - (κ / N^{2}) b \nabla b$ and dissipation $ν ∣ \nabla u ∣^{2} + (κ / N^{2}) ∣ \nabla b ∣^{2}$ .

3.1givenA1A2F1F2L1L2L3L4L6step 1.2step 2.1algebra∎

Write $e = \frac{1}{2} (∣ u ∣^{2} + b^{2} / N^{2})$ , $F = π u - ν \sum_{i = 1}^{3} u_{i} \nabla u_{i} - (κ / N^{2}) b \nabla b$ , and $D = ν ∣ \nabla u ∣^{2} + (κ / N^{2}) ∣ \nabla b ∣^{2}$ , so step 2.1 gives $\partial_{t} e + \nabla \cdot F = - D$ . Fix a time $t$ . Each $\partial_{x_{j}} F_{j} (t, \cdot)$ is continuous on $Q_{3}$ ; [L4] in $x_{j}$ gives its integral as the periodic endpoint difference $F_{j} ∣_{x_{j} = 2 π} - F_{j} ∣_{x_{j} = 0} = 0$ . Applying [L1], [L3], and [L6] only in space gives $R_{Q_{3}} (\nabla \cdot F (t, \cdot)) = 0$ , hence $R_{Q_{3}} (\partial_{t} e (t, \cdot)) = - R_{Q_{3}} (D (t, \cdot))$ . Integrate this last equality in time; step 1.2 makes both section-integral functions Riemann integrable, and [L3] gives $R_{K} (\partial_{t} e) = - R_{K} (D)$ . For each fixed $x$ , [L4] in time applied to $e$ and then [L3] yield $R_{K} (\partial_{t} e) = R_{Q_{3}} (e (t_{1}, \cdot) - e (t_{0}, \cdot))$ . By [L6] and [L2] at the endpoints, this is $E (t_{1}) - E (t_{0})$ ; by step 1.2, $R_{K} (D)$ is the displayed nested Lebesgue dissipation integral. No time integral of an individual pressure-flux derivative is used. Since $N > 0$ , $E (t) \geq 0$ ; since $ν, κ \geq 0$ , the dissipation is nonnegative, and when $ν = κ = 0$ it vanishes.

Depends on

Dependency tree · 2 levels

Sources

Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, University of Exeter lecture notes, Autumn 2018, §1.6.3, equations (1.72)–(1.78)