r = L_{f} / L_{ρ}

fixed in

(0, 1)

. Then

\frac{K _{\infty}}{U _{0}^{2}} = \frac{L _{v}}{L _{ρ}} \frac{1 - r}{r}, \frac{B}{ϵ} = \frac{L _{v}}{L _{ρ}} \frac{U _{0}^{2}}{K} .

(1)

For

L_{v} ≪ L_{ρ}

, dissipation is strong at a given

K

and

K_{\infty} ≪ U_{0}^{2}

when

r

is not extreme. If the initial energy is of order

U_{0}^{2}

, it decays toward a small steady value and starts with a small instantaneous mixing efficiency. This is a regime of short velocity-fluctuation scales and efficient viscous loss relative to the modeled buoyancy conversion at that initial energy.

For

L_{v} ≫ L_{ρ}

, dissipation is weak at that same energy and

K_{\infty} ≫ U_{0}^{2}

. A moderate initial energy grows and initially has a large instantaneous mixing efficiency; increasing stored energy eventually makes dissipation comparable with the remaining forcing. The energy-relaxation time is of order

L_{v} / K_{\infty}

, which grows as

L_{v}

at fixed forcing and density lengths.

Within this closure, both limits have the same long-time loss fraction $η_{c, \infty} = L_{f} / L_{ρ}$ . The steady energy changes with

L_{v}

so as to compensate for its explicit effect on dissipation. Different initial conditions and finite observation windows give different transient efficiencies; one cannot infer a universal long-time efficiency from