Shallow moment equations

SME resolves the vertical profile of the horizontal velocity of a shallow free-surface flow as a polynomial: the depth mean plus level moments. level=0 is shallow water. Paper: Kowalski and Torrilhon 2019, Moment approximations and model cascades for shallow flow.

Starting point

The reduced balance laws for velocity \((u, v, w)\), pressure \(p\) and deviatoric stress \(\sigma\), constant density \(\rho\), gravity \(g(e_x, e_y, e_z)\), bed \(h_b\) and free surface \(h_s\):

\[ \partial_x u + \partial_y v + \partial_z w = 0 \]

\[ \partial_t u + \partial_x (u^2) + \partial_y (uv) + \partial_z (uw) = -\frac{1}{\rho} \partial_x p + \frac{1}{\rho} \partial_z \sigma_{xz} + g e_x \]

\[ \partial_t v + \partial_x (uv) + \partial_y (v^2) + \partial_z (vw) = -\frac{1}{\rho} \partial_y p + \frac{1}{\rho} \partial_z \sigma_{yz} + g e_y \]

\[ p(t,x,y,z) = (h_s(t,x,y) - z) \rho g e_z \]

with the kinematic conditions at the free surface and at the bed

\[ \partial_t h_s + (u(t,x,y,h_s), v(t,x,y,h_s))^T \cdot \nabla h_s = w(t,x,y,h_s) \]

\[ \partial_t h_b + (u(t,x,y,h_b), v(t,x,y,h_b))^T \cdot \nabla h_b = w(t,x,y,h_b) \]

the Newtonian shear stress, the stress-free surface and the slip condition at the bed with slip length \(\lambda\)

\[ \sigma_{xz} = \mu \partial_z u, \qquad \partial_z u|_{z=h_s} = 0, \qquad (u - \lambda \partial_z u)|_{z=h_b} = 0 \]

and the same for \(v\). In the mapped coordinate \(\zeta \in [0, 1]\) the velocity is the mean plus a polynomial deviation

\[ \zeta = \frac{z - h_b(t, x, y)}{h(t, x, y)}, \qquad u(x,y,t,\zeta) = u_m(x,y,t) + \sum_{j=1}^{N} \alpha_j(x,y,t) \phi_j(\zeta) \]

with \(h = h_s - h_b\) and the scaled Legendre polynomials \(\phi_j\), orthogonal on \([0, 1]\) with \(\phi_j(0) = 1\). The moment equations are the momentum balance tested with \(\phi_i\) and integrated over \(\zeta\).

Assumptions

• Shallow flow: the in-plane deviatoric stresses \(\sigma_{xx}\), \(\sigma_{yy}\), \(\sigma_{xy}\) are neglected.

• Hydrostatic pressure (3.4), constant density.

• Newtonian shear stress, stress-free surface, Navier slip at the bed, enforced weakly through the integrated stress terms.

• Velocity profile: a polynomial of degree \(N\) in \(\zeta\) (4.3).

Generated equations

SME(level=1, dimension=2, closures=[Newtonian(), NavierSlip(), StressFree()])

State \(Q\): \(b\), \(h\), \(q_{0}\), \(q_{1}\)

Parameters: g = 9.81, rho = 1.0, nu = 0.0, lambda_s = 0.0, e_x = 0.0

bottom

\[ \partial_{t} b = 0 \]

mass_0

\[ \partial_{t} h + \partial_{x} q_{0} = 0 \]

momentum_x_0

\[\begin{split} \begin{aligned} &\partial_{t} q_{0} + \partial_{x} \left(\frac{q_{0}^{2}}{h}\right) + \partial_{x} \left(\frac{g h^{2}}{2}\right) + \partial_{x} \left(\frac{q_{1}^{2}}{3 h}\right) + g h \, \partial_{x} b \\ &\quad + \frac{\lambda_{s} q_{0}}{\rho h} - \frac{\lambda_{s} q_{1}}{\rho h} - e_{x} g h = 0 \end{aligned} \end{split}\]

momentum_x_1

\[\begin{split} \begin{aligned} &\partial_{t} q_{1} + 2 \, \partial_{x} \left(\frac{q_{0} q_{1}}{h}\right) - \frac{q_{0}}{h} \, \partial_{x} q_{1} - \frac{3 \lambda_{s} q_{0}}{\rho h} + \frac{3 \lambda_{s} q_{1}}{\rho h} \\ &\quad + \frac{12 \nu q_{1}}{h^{2}} = 0 \end{aligned} \end{split}\]

Notation: \(q_0 = h u_m\) and \(q_k = (-1)^k h \alpha_k\) (Zoomy’s basis has \(\phi_k(1) = 1\)), \(\lambda_s = \rho \nu / \lambda\), \(b = h_b\), \(e_z = 1\), level \(= N\). The row momentum_x_k is \((-1)^k\) times the paper’s equation for \(\alpha_k\). With a flat bed and \(e_x = 0\), level=0 is the paper’s (4.13) and level=1 its (4.14) with (4.15). tests/model/test_model_references.py in zoomy_core checks levels 1 to 3 row by row against (4.7) and (4.10).

Options

setting
level	moments above the mean, 0 is shallow water
dimension	2: (x, z), state b, h, q_0, ..., q_level. 3: (x, y, z), state b, h, q_x_k, q_y_k
closures	for example [Newtonian(), NavierSlip(), StressFree()]. Empty leaves the shear stress unclosed
stress	Stress.shallow keeps the vertical shear (the paper’s model), Stress.full the whole Newtonian tensor
scaling	a ShallowScaling: the regime of the stress truncation
small_slope	True shallow boundary tractions, False slope-aware ones
quadrature_order	Gauss-Legendre order for non-polynomial closures, 0 off
parameters	g, rho, nu, lambda_s, e_x

from zoomy_core.model.models import SME, Newtonian, NavierSlip, StressFree
from zoomy_core.systemmodel import SystemModel

model = SME(level=1, dimension=2, closures=[Newtonian(), NavierSlip(), StressFree()],
            parameters={"nu": 0.1, "lambda_s": 0.5})
sm = SystemModel.from_model(model)

In the GUI

Compose, Model card Shallow Moments: set level in its parameters panel. Pick a mesh and the Hyperbolic (NumPy) solver and run. The card writes SME(...) with [Newtonian(), NavierSlip(), StressFree()] into the ## Model section of case.py.