A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid
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Journal of Mathematical Fluid Mechanics
A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid Prince Romeo Mensah Communicated by E. Feireisl
Abstract. We study a singular limit of a scaled compressible Navier–Stokes–Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say ε 1, whereas the Rossby number scales like εm for m > 1 large, then we show that any family of weak martingale solution to the 3-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as ε → 0, to the 2-D incompressible Navier–Stokes system with a corresponding random forcing term. Mathematics Subject Classification. Primary 35R60, 35Q35; Secondary 76M45, 76N99. Keywords. Stochastic compressible fluid, Navier–Stokes–Coriolis, Martingale solution, Mach number, Rossby number, Froude number.
1. Introduction Our aim is to study the following singular limit problem for rotating fluids d + div(u)dt = 0, 1 1 (e3 × u) + ∇p() dt d(u) + div(u ⊗ u) + Ro Ma2 1 = div S(∇u) dt + 2 ∇G + Φ(, u)dW Fr
(1.1)
where the density and velocity vector field u takes its values from the space O = R2 × (0, 1). The term 1 1 Ro (e3 × u) in (1.1) above accounts for rotation in the fluid due to Coriolis forces and the factor Ro - which is the reciprocal of the Rossby number - measures the intensity or the speed of this rotation. e3 = (0, 0, 1) is the unit vector in the vertical x3 -direction. The centrifugal force term is essentially of the form ∇G ≈ ∇(|x1 |2 + |x2 |2 ) with (x1 , x2 ) ∈ R2 and with Fr12 - the squared reciprocal of the Froude number - quantifying the level of stratification in the fluid. Here, p() = γ with γ > 32 is the isentropic pressure, Ma is the Mach number and the viscous stress tensor is 2 (1.2) S(∇u) := ν ∇u + ∇T u − div uI 3 with viscosity coefficient satisfying ν > 0 . As would be made clear in Sect. 3.1, we intend to use a version of Korn’s inequality and as such, it is of technical importance to omit the bulk part of the viscous stress tensor. The author acknowledges the financial support of the Department of Mathematics, Heriot-Watt University, through the James–Watt scholarship. 0123456789().: V,-vol
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P. R. Mensah
JMFM
A prototype for the stochastic forcing term will be Φ(, u) dW ≈ dW 1 + u dW 2
(1.3)
for a pair of independent identically distributed Wiener processes W 1 and W 2 . We give the precise assumptions on the noise term in Sect. 2.2. If we set the Rossby number Ro = ε, the Froude number Fr = ε and the Mach number Ma = εm for some m 1, then given a sequence (ε , uε ) of weak martingale solutions to (1.1) (see Definition 2.2 for the precise definition), we show that its limit U = [Uh (x1 , x2 ), 0], solves the 2-dimensional Navier–Stokes system dUh + [divh (Uh ⊗ Uh ) + ∇h π − νΔh Uh ] dt = PΦ(1, Uh ) dW, divh Uh = 0.
(1.4)
Here, π is an associated pressure term, P represen
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