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A force density is force per unit volume, measured in $\mathrm{N}{\mathrm{m}}^{-3}$. Pressure gradients, the divergence of a viscous stress and gravitational body force have this dimension in a local momentum balance.

For a central differentiable pair potential, an exact local representation of the interaction momentum flux isHere $n_2$ is the two-particle number density. The line integral assigns each pair's exchanged momentum along the segment joining the particles. Translating both endpoints gives ${\partial }_s{n}_2=\mathbf{R}\cdot {\nabla }_{\mathbf{x}}{n}_2$. Integrating over $s$ and using pair-exchange symmetry proves that $-\nabla \cdot {\mathbf{P}}^{\varphi }$ is the interaction force density. Its homogeneous isotropic specialization gives the interaction part of the virial pressure from pair correlations. The kinetic pressure tensor accounts separately for random particle motion.

Use the source's Gaussian units and neglect the displacement current, as in nonrelativistic magnetohydrodynamics. Write $\Delta p=p_{\perp }-p_{\parallel }$ for the pressure anisotropy. The pressure tensor is $\mathsf{P}=p_{\perp }\mathsf{I}-\Delta p\mathbf{bb}$, so its force density isAmpère's law and the solenoidal magnetic-field constraint give the Lorentz force decompositionAdding these two force densities and using $\mathbf{BB}=B^2\mathbf{bb}$ yieldsThe first term is the gradient of the modified magnetohydrodynamic total pressure; the second is the anisotropic Maxwell stress tensor contribution together with the directional pressure force. Their effective coefficient is the pressure-anisotropy-modified magnetic tension, $\mathcal{T}=B^2/(4\pi )+\Delta p$. At fixed magnetic field, positive pressure anisotropy increases resistance to bending, while negative pressure anisotropy reduces it. When $\mathcal{T}$ becomes negative, bending experiences an amplifying force rather than a restoring force. Gradients of $\Delta p$ can also drive motion, so the curvature interpretation refers specifically to the bending contribution.

For fixed $\mathbf{R}$ and $t$, define $h(\mathbf{x},s)=n_2(\mathbf{a},\mathbf{b},t)$ with $\mathbf{a}=\mathbf{x}+(s-1)\mathbf{R}$ and $\mathbf{b}=\mathbf{x}+s\mathbf{R}$. The chain rule givesso $\boxed{{\partial }_{s}h=\mathbf{R}\cdot {\nabla }_{\mathbf{x}}h}$. The derivative with respect to $\mathbf{x}$ here translates both arguments of the two-particle number density; it is not a derivative of its first argument alone.

Take the spatial divergence of the proposed interaction pressure tensor. Its pair potential factor is independent of $\mathbf{x}$, so the preceding identity givesExchange symmetry of the two-particle distribution function implies $n_2(\mathbf{x}-\mathbf{R},\mathbf{x})=n_2(\mathbf{x},\mathbf{x}-\mathbf{R})$. In the second term, substitute $\mathbf{R}\mapsto -\mathbf{R}$. Since $R_i{\varphi }^{\prime }(R)/R$ is odd, both terms become equal:On the other hand, at fixed ${\mathbf{x}}_{\star }=\mathbf{x}+\mathbf{R}$, the first-argument derivative of the pair potential is ${\nabla }_{\mathbf{x}}\varphi =-\mathbf{R}{\varphi }^{\prime }(R)/R$. Thus the interaction force density in b is $-{\partial }_{x_j}{P}_{ij}^{\varphi }$. Moving it to the left yields local conservation of momentum:The integral over $s$ distributes each pair's momentum flux along the straight segment joining the particles. The factor $1/2$ removes the double counting of ordered pairs. Central forces also make $P_{ij}^{\varphi }=P_{ji}^{\varphi }$.

Use the BGK model from b. Multiply it by $v_i$ and integrate over velocity space. Matching the collision invariants makes the collision contribution vanish. For the external force term, integration by parts givesDefine the momentum flux and force density byThe resulting momentum conservation equation isThe pressure tensor is the second central velocity moment,The zeroth velocity moment also supplies the continuity equation ${\partial }_t\rho +{\partial }_j(\rho u_j)=0$, which will be needed in d.

Resolve gravity into components $g\sin \alpha$ down the bed and $-g\cos \alpha$ normal to it. For slow incompressible flow, inertia is negligible and Cauchy momentum equation reduces to the divergence of the stress tensor plus the gravitational force density being zero. Write the dominant shear component as $\tau ={\sigma }_{xz}$ and the leading normal stresses as $-p$.

The shallow-ice approximation assumes a small thickness-to-length ratio. Vertical variation of the shear is retained in the along-bed equation, while longitudinal deviatoric-stress gradients and the shear contribution to the normal equation are higher order. The leading balances are consequentlyThe second is a hydrostatic pressure balance; the first balances gravitational driving and the surface-induced pressure gradient against vertical shear. This approximation requires shallow geometry and sufficiently slow flow, not a Newtonian constitutive equation for ice.

Subtract the background hydrostatic pressure. A relief amplitude $h$ produces a restoring pressure perturbation $p^{\prime }\sim \rho gh$ and horizontal gradient $\rho gh/\lambda$. Incompressibility makes the disturbance penetrate a depth of order its horizontal scale. For sinusoidal viscous rebound, with relaxation speed $v\sim h/\tau$, the viscous force density is $\eta v/{\lambda }^2$. Balancing givesThis is a small-amplitude viscous half-space estimate; numerical factors from the sinusoidal geometry are absent from a scaling argument.

Unsteady inertial force density scales as $\rho v/\tau$, whereas viscous force density scales as $\eta v/{\lambda }^2$. Using the preceding timescale givesConvective inertia has ratio $\rho v\lambda /\eta$. Since $v\sim h/\tau$, this is smaller by $h/\lambda$ for small relief. Thus the displayed estimate controls the larger inertial contribution in this model.

Consider adiabatic perturbations of a static ideal gas, with background pressure $p_0(\mathbf{x})$, mass density ${\rho }_0(\mathbf{x})$ and constant heat-capacity ratio $\gamma$. Assume a prescribed force density $-\nabla \chi$ that does not change when the fluid is perturbed. Static balance is $\nabla (p_0+\chi )=0$. Linearized Euler equations and adiabatic compression then giveDifferentiating the second equation and substituting the first yieldsThe divergence form gives weighted acoustic Green-function reciprocity. Neither uniform mass density nor uniform entropy is required. A prescribed gravitational acceleration instead produces a force density proportional to the perturbed mass density; its extra buoyancy term cannot be omitted under this derivation. The homogeneous limit is the acoustic pressure wave equation.