Covariant Expressions for Vector and Tensor Operators

This page summarizes the covariant expressions for vector and tensor operators used in Athena++. These operators are mostly used for computing viscosity and thermal conduction terms in Diffusion Processes. Please see Appendix A of Stone & Norman (1992) for details.

The metric tensor for orthogonal coordinate systems reads

where h₁, h₂, and h₃ are scale factors or Lamé coefficients. They take different forms depending on the specific coordinate system (see Table 1 below).

Table 1: Scale factors and their derivatives in various coordinate systems.

scale factors	Cartesian	Cylindrical	Spherical
h1	1	1	1
h2	1	r	r
h3 = h31 · h32	1	1	r · sinθ
h31	1	1	r
h32	1	1	sinθ
∂h2 / ∂x1	0	1	1
∂h31 / ∂x1	0	0	1
∂h32 / ∂x2	0	0	cosθ

Gradient of a scalar:

∇ Φ = ∂Φ / (h_i ∂x_i) ê_i ,

with i ∈ [x1,x2,x3] and assuming Einstein summation (the repeated indices are implicitly summed over).

Divergence of a vector:

∇ · F = g^-½ ∂(g^½ F_i / h_i) / ∂x_i,

where F = F_i ê_i , and g^½ = h₁h₂h₃.

Curl of a vector:

∇ × F = h_k g^-½; ϵ_kji ∂(h_i F_i) / ∂x_j ê_i.

Gradient of a vector (covariant derivative):

∇ F = [∂(h_iF_i) / ∂x_j - Γ^k_{i j} h_k F_k] (h_i h_j)^-1 ê_i ê_j ,

where Γ^k_{i j} is the Christoffel symbol:

Γⁱ_{i i} = (2h_i²)^-1 ∂ h_i² / ∂ x_i

Γⁱ_{i j} = Γⁱ_{j i} = (2h_i²)^-1 ∂ h_i² / ∂ x_j

Γⁱ_{j j} = -(2h_i²)^-1 ∂ h_j² / ∂ x_i

Γⁱ_{j k} = 0

Example

For instance, the viscous stress tensor (neglecting the bulk viscosity) can be written as

σ_{i j} = -μ( v_{i; j} + v_{j; i} -⅔ v^k_;k g_{i j} ),

where μ = ρ ν is the dynamic viscosity. The expression consists of two gradient terms and one term containing the divergence of the velocity. All of these terms can be calculated in any of the supported Coordinate Systems and Meshes using the above formulae.