Kinematic boundary conditions for fluid dynamics in spherical geometry | Geowanderer

Stress-free boundary condition

The stress tensor for Newtonian fluid is given by

\[\sigma = \rho \nu \dot{\varepsilon}\]

where \(\dot{\varepsilon}\) is the strain rate tensor given by

\[\dot{\varepsilon} = \nabla \mathbf{u} + \nabla \mathbf{u}^\top - \frac{2}{3}(\nabla\cdot\mathbf{u})\mathbf{I}.\]

Stress-free boundary condition dictates that the tangential components of the surface traction vanishes, i.e. \(\hat{\mathbf{n}}\times (\hat{\mathbf{n}} \cdot \sigma)\vert_{\partial V} = \mathbf{0}\); equivalently, this means that the tangential components of the surface strain rate vanishes, i.e. \(\hat{\mathbf{n}}\times (\hat{\mathbf{n}} \cdot \dot{\varepsilon})\vert_{\partial V} = \mathbf{0}\). Here \(\hat{\mathbf{n}}\) is the surface normal and is equal to \(\hat{\mathbf{r}}\) in spherical geometry. Therefore, the condition on the stress can be transferred to the strain rate, dictating that the tangential components of the normal directional strain rate (\(\hat{\mathbf{n}}\cdot \dot{\varepsilon}\)) vanishes on the surface. In terms of velocity, the normal directional strain rate takes the form

\[\begin{aligned} &\hat{\mathbf{n}}\cdot \dot{\varepsilon} = \hat{\mathbf{r}}\cdot \dot{\varepsilon} \\ &= \hat{\mathbf{r}} \cdot \left[\nabla \mathbf{u} + \nabla \mathbf{u}^\top - \frac{2}{3}(\nabla\cdot\mathbf{u})\mathbf{I} \right] \\ &= \partial_r \mathbf{u} + \nabla \mathbf{u} \cdot \hat{\mathbf{r}} - \frac{2}{3}(\nabla\cdot\mathbf{u})\hat{\mathbf{r}} \\ &= (\partial_r u_r \hat{\mathbf{r}} + \partial_r \mathbf{u}_H) + \nabla (\mathbf{u} \cdot \hat{\mathbf{r}}) - \nabla \hat{\mathbf{r}} \cdot \mathbf{u} - \frac{2}{3}(\nabla\cdot\mathbf{u})\hat{\mathbf{r}} \\ &= (\partial_r u_r \hat{\mathbf{r}} + \partial_r \mathbf{u}_H) + \nabla u_r - \frac{1}{r}(\mathbf{I} - \hat{\mathbf{r}}\hat{\mathbf{r}}) \cdot \mathbf{u} - \frac{2}{3}(\nabla\cdot\mathbf{u})\hat{\mathbf{r}} \\ &= (\partial_r u_r \hat{\mathbf{r}} + \partial_r \mathbf{u}_H) + \nabla u_r - \frac{\mathbf{u} - u_r \hat{\mathbf{r}}}{r} - \frac{2}{3}(\nabla\cdot\mathbf{u})\hat{\mathbf{r}} \\ &= \hat{\mathbf{r}} \left[2\partial_r u_r - \frac{2}{3}(\nabla\cdot\mathbf{u})\right] + \left[\partial_r \mathbf{u}_H + \nabla_H u_r - \frac{\mathbf{u}_H}{r}\right] \end{aligned}\]

where the radial / surface-normal component and the horizontal / surface-tangent components have been separated, and the subscript \(H\) indicates horizontal component / differential. The radial part is unconstrained, where the surface-tangent components need to vanish at the surface, hence

\[\left(\partial_r \mathbf{u}_H + \nabla_H u_r - \frac{\mathbf{u}_H}{r}\right)\bigg\vert_{\partial V} = \mathbf{0}\]

Under no-penetration boundary condition (which is almost always the case in simulations in spherical geometry), we have \((\nabla_H u_r)\vert_{\partial V} = \nabla_H u_r\vert_{\partial V} = \mathbf{0}\). The kinematic condition therefore simplifies to

\[\begin{aligned} \left(\partial_r \mathbf{u}_H - \frac{\mathbf{u}_H}{r}\right)\bigg\vert_{\partial V} &= \mathbf{0}\\ \text{or}\quad\frac{\partial}{\partial r} \left(\frac{\mathbf{u}_H}{r}\right)\bigg\vert_{\partial V} &= \mathbf{0}. \end{aligned}\]

Tor-Pol representation, incompressible

Under incompressible approximation, the solenoidal velocity field can be represented using one toroidal and one poloidal scalar, via

\[\begin{aligned} \mathbf{u} &= \nabla\times T\mathbf{r} + \nabla\times \nabla\times P \mathbf{r} \\ &= \nabla_s T \times \hat{\mathbf{r}} + \nabla_s \frac{1}{r}\partial_r(rP) - \hat{\mathbf{r}} \nabla_s^2 \frac{P}{r} \end{aligned}\]

where \(\nabla_s\) is the angular gradient (the dimensionless form of the horizontal gradient \(\nabla_H\), i.e. \(\nabla_s = r \nabla_H\)), and \(\nabla_s^2\) is the angular Laplacian. Substituting the horizontal components in the stress-free boundary condition,

\[\partial_r \left(\frac{\mathbf{u}_H}{r}\right)\bigg\vert_{\partial V} = \nabla_s \partial_r\left(\frac{T}{r}\right)\bigg\vert_{\partial V} \times \hat{\mathbf{r}} + \nabla_s \partial_r\left[\frac{1}{r^2}\partial_r(rP)\right]\bigg\vert_{\partial V} = \mathbf{0}\]

Considering that toroidal and poloidal components are linearly independent function spaces on the surface of the sphere, and the uniqueness of representing a surface-tangent vector field in toroidal and poloidal scalars (up to some constant) (this is equivalently the Helmholtz decomposition theorem on a surface), we can write

\[\left\{\begin{aligned} \frac{\partial}{\partial r}\left(\frac{T}{r}\right)\bigg\vert_{\partial V} = \frac{1}{r} \left(\frac{\partial T}{\partial r} - \frac{T}{r}\right)\bigg\vert_{\partial V} = 0 \\ \frac{\partial}{\partial r}\left[\frac{1}{r^2}\frac{\partial}{\partial r}(rP)\right]\bigg\vert_{\partial V} = \frac{1}{r} \left(\frac{\partial^2 P}{\partial r^2} - \frac{2P}{r^2}\right)\bigg\vert_{\partial V} = 0 \end{aligned}\right.\]

Given that no-penetration condition requires \(P\vert_{\partial V} = 0\), the poloidal condition is simply

\[P\vert_{\partial V} = 0 \quad \& \quad \partial_r^2 P\vert_{\partial V} = 0.\]

Tor-Pol representation, anelastic

The boundary condition for toroidal poloidal scalars in the anelastic approximation is very similar, just a bit more complicated. In anelastic approximation, since \(\nabla\cdot (\bar{\rho} \mathbf{u}) = 0\), the mass flux instead of the flow is solenoidal, and can be represented via the toroidal-poloidal decomposition

\[\bar{\rho}\mathbf{u} = \nabla_s T \times \hat{\mathbf{r}} + \nabla_s \frac{1}{r}\partial_r(rP) - \hat{\mathbf{r}} \nabla_s^2 \frac{P}{r}\]

Assuming that the background density profile \(\bar{\rho}\) is simply a function of \(r\), i.e. background has spherical symmetry, multiplication and division of \(\bar{\rho}\) commute with all angular operators. The stress-free boundary condition yields

\[\partial_r \left(\frac{\mathbf{u}_H}{r}\right)\bigg\vert_{\partial V} = \nabla_s \partial_r\left(\frac{T}{\bar{\rho}r}\right)\bigg\vert_{\partial V} \times \hat{\mathbf{r}} + \nabla_s \partial_r\left[\frac{1}{\bar{\rho}r^2}\partial_r(rP)\right]\bigg\vert_{\partial V} = \mathbf{0},\]

which in turn yields the toroidal condition

\[\frac{\partial}{\partial r}\left(\frac{T}{\bar{\rho}r}\right)\bigg\vert_{\partial V} = 0\]

and the poloidal condition

\[\begin{aligned} \frac{\partial}{\partial r}\left[\frac{1}{\bar{\rho}r^2}\frac{\partial}{\partial r}(rP)\right]\bigg\vert_{\partial V} = 0 \\ \frac{\partial}{\partial r}\left[\frac{\partial_r P}{\bar{\rho}r} + \frac{P}{\bar{\rho}r^2} \right]\bigg\vert_{\partial V} = 0 \\ \left[\frac{\partial_r^2 P}{\bar{\rho}r} + \frac{\partial_r P}{r} \partial_r \frac{1}{\bar{\rho}} + P \partial_r \frac{1}{\bar{\rho} r^2}\right]_{\partial V} = 0 \end{aligned}\]

Again, if under no-penetration condition, \(P\vert_{\partial V} = 0\), then the poloidal condition for stress-free boundary reduces to

\[\left[\frac{\partial_r^2 P}{\bar{\rho}r} + \frac{\partial_r P}{r} \partial_r \frac{1}{\bar{\rho}}\right]_{\partial V} = \frac{1}{r} \frac{\partial}{\partial r} \left(\frac{\partial_r P}{\bar{\rho}}\right)\bigg\vert_{\partial V} = 0\]

We also see that condition re\verts back to \(\partial_r^2 P\vert_{\partial V} = 0\) when \(\bar{\rho} = \text{Cst}.\), which is the case under the incompressible approximation.