The Hodge-Helmholtz decomposition of a vector field, ~u?, is
~u?=∇ ×A~+∇φ.
For our purposes, we do not wish to construct a discrete curl,C, that mimics the properties of its continuous counterpart, DC≡0,CG≡0, and C≡ −CT. To do so would potentially require a major overhaul of our divergence and gradient discretizations, which would in turn effect the freestream preserving construction of our metric elements. Instead, we rely on an alternate form of the decomposition,
~
u?=~u+∇φ ∇ ·~u= 0,
(2.15)
where ~uis by definition a soleniodal, or divergence-free vector field. It is worth pointing out that the only reason we do not need a mimetic C is because the curl does not show up in our formulation of the dynamical equations 1.20.
The discrete version of eq. 2.15 is
~u?=~u+Gφ
D~u= 0,
(2.16)
where~uis now understood to be defined on the grid’s face-centers and weighted byJ. This equation mimics all of the necessary properties of the continuous Hodge-Helmholtz decomposition. First,~uandGφbelong to orthogonal vector spaces since
hGφ, ~ui=− hφ,D~ui= 0,
meaning the solenoidal and irrotational terms of the decomposition can be thought of as orthogonal compo- nents of~u?. Second, the energy put into a system by a timestepping algorithm can be written as
1 2h~u ?, ~u?i=1 2h~u, ~ui+ 1 2hGφ,Gφi.
Therefore, if we remove the irrotational component,Gφ, we will not remove or alter the component of the energy that owes itself to the dynamical velocity. Third, we can remove the irrotational component of~u? by solving Poisson’s equation,
DGφ=D~u? ~u=~u?−Gφ.
For brevity, this operation is called avelocity projection.
The idempotent operatorP= 1−G(DG)−1Dis a projection operator. It exacly removes the irrotational component of vectors in the staggered space Gm0 . All cell-centered quantities (buoyancy and momentum) will be advected using an advecting velocity, uAD∈ Gm0 , that has been projected in this exact manner. However, since our advection scheme is a finite volume method, we must update a cell-averaged velocity. (Since cell-averaged quantities are also cell-centered quantities to second order in the grid spacing, we will simply call the cell-averaged velocity the CC velocity.) This requires non-staggered versions of the gradient and divergence operators. We will call these the CC gradient and CC divergence operators. To construct these operators, we introduce two averaging operators that change the centering of grid data, AvFC→CCand AvCC→FC. These are defined componentwise via
AvFC→CC[u]i,j,k=1 2 ui+1/2,j,k−ui−1/2,j,k AvFC→CC[v]i,j,k= 1 2 vi,j+1/2,k−vi,j−1/2,k AvFC→CC[w]i,j,k= 1 2 wi,j,k+1/2−wi,j,k−1/2 and AvCC→FC[u]i+1/2,j,k=1 2 ui+1,j,k−ui−1,j,k AvCC→FC[v]i,j+1/2,k=1 2 vi,j+1,k−vi,j−1,k AvCC→FC[w]i,j,k+1/2= 1 2 wi,j,k+1−wi,j,k−1 ,
where ~u=uξˆ+vηˆ+wζˆ and the centerings are implicitly declared by the cell/face indices. With these averaging operators, the CC gradient of a CC scalar, φ, and the CC divergence of a CC velocity, ~u are defined by
GCCφ= AvFC→CC[Gφ]
DCC~u=DAvCC→FC[~u].
Our CC projector is now defined byPCC= 1−GCC(DG)−1DCC. This is not idempontent since (DG)−1 is not the inverse of DCCGCC. The CC projector is therefore only an approximate projector and does not completely remove the irrotational component of a CC vector field in one application. It has been shown that the composite operator (PCC)n tends towards becoming an exact projector in the limitn→ ∞ and
that this limit is well defined and stable, but only one application ofPCC is sufficient to produce a stable timestepping algorithm [49].
Upon application of the CC projector to a velocity field, we see that by definition of DCC,
PCC~u=~u−GCC(DG)−1DCC~u
=~u−(various operations) AvCC→FC[~u].
In words, when we project a CC velocity field, ~u, we are actually removing the irrotational component of the filtered velocity, AvCC→FC[~u]. The filtered modes that cannot be removed from the velocity are of the highest frequency supported by the grid and appear as checkerboards or stripes of covergent/divergent fluid. Attempts at filtering these modes with a Shapiro filter [52] and a compact, 4th-order filter derived by Lele [53] have proven to be an unstable detriment to the dynamics. A truly mimetic discretization may prove to be a more fruitful option, since it would preclude the need for FC and CC sets of velocities while reducing the stencil size of the Laplacian operator. But these efforts have not yet been explored for use in SOMAR. We find that simply providing adequate resolution at the base level (the coarsest AMR level) reduces the high frequency modes provided to the CC projector in the first place.
As a final note, attempts have been made to construct an idempotent, exact CC projector, but without much success. The simplest approach would be to use all CC operators in the projector’s construction,
PCC= 1−GCC(DCCGCC)−1DCC. This, however, checkerboards the grid due to the wide stencil of the CC Laplacian,DCCGCC, again promoting errors of the highest frequencies supported by the grid.