Abstract
This article is devoted to providing a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable “mixing” conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.
Appendix
Proof of Lemma 4
A similar result has been given in [Citation11, Proposition 7]. Here we present another proof which gives a stronger convergence, together with the convergence rate in probability.
First, under the assumption (Hg), we show that for almost all ω ∈ Ω,
(A.1) for every q ∈ H.
Noticing
we get
Consider
By a mixing property [Citation4, Proposition 7.2.2 ], we have
(A.2) Noticing that
by estimate (Equation43
(A.2) ) we have (Equation42
(A.1) ).
By the estimate in Section 2.1, for every κ > 0, there is CκT > 0, which is independent of ε, such that
(A.3) for every t ⩾ s ⩾ 0. Furthermore, by the tightness of the distributions of {uε} in space C(0, T; H), for every κ > 0, there is a compact set Kκ⊂C(0, T; H) such that
(A.4) So we define
Due to the compactness of Kκ, for every ϵ > 0, we only need to consider a finite ϵ-net {q1 , q2 , … , qN} in C(0, T; H), which covers {uε}. Without loss of generality, we assume that qj, j = 1, 2, …, N, are simple functions [Citation11].
Now we consider all ω ∈ Ωκ. By the construction of and boundedness of f, we have for ω ∈ Ωκ
for some constant C > 0.
For every δ > 0, we partition the interval [0, T] into subintervals of length of δ. Then for t ∈ [kδ, (k + 1)δ), ,
for some qj. Notice that, by the assumption (Hg) and the definition of
,
is also Lipschitz continuous in u with the same Lipschitz constant Lg. Then by the assumption (Hg) and the definition of Ωκ,
Due to the arbitrary choice of δ, ϵ, and κ, and notice a similar discussion as that in [Citation11], we thus complete the proof.