# 213. Variable-Background Emergent Hydrodynamic Form Appendix 207 derived the emergent hydrodynamic form in an approximately uniform region, where the local transport scale $k$ could be treated as constant. This appendix extends that derivation to the case in which $$ k=k(\mathbf{r},t) $$ varies across the resolved continuum. The balance-law structure derived here is exact once two variable-background ingredients are adopted: - the matched constitutive relation $$ \mathbf{g}=\frac{\mathbf{S}}{k^2}, $$ - the residual background momentum-exchange term $$ \mathbf{f}_{\mathrm{bg}}. $$ Appendix 214 derives the first of these exactly inside the symmetric constitutive closure used in the gravity chapters. What is not yet derived here is the full substrate-specific closure that fixes both objects from the underlying transport. The novelty is that once $k$ varies, one must distinguish carefully between: - conserved coarse-grained energy, - effective inertial density, - momentum exchange with the varying background. ## 213.1 Local Variables in a Varying Background Let $$ \beta(\mathbf{r},t):=\frac{1}{k(\mathbf{r},t)^2}. $$ At the resolved scale, keep the local energy density and energy flux $$ u, \qquad \mathbf{S}, $$ with exact local energy continuity $$ \partial_t u+\nabla\cdot\mathbf{S}=0. $$ Adopt, as the variable-background extension of the uniform-region relation, the local momentum density $$ \mathbf{g}:=\beta\,\mathbf{S}=\frac{\mathbf{S}}{k^2}. $$ When the background varies, the resolved subsystem need not be momentum-closed by itself. Define the exact background-exchange density $$ \mathbf{f}_{\mathrm{bg}}(\mathbf{r},t) $$ by the local balance law $$ \partial_t\mathbf{g}-\nabla\cdot\mathbf{T} = \mathbf{f}_{\mathrm{bg}}. $$ This is not a new force law. It is the exact residual bookkeeping term once $\mathbf{g}$ and $\mathbf{T}$ are specified: whatever momentum is not balanced by the resolved stress transport is, by definition, momentum exchanged with the unresolved background constitutive organization. In a uniform region, $$ \nabla k=0, \qquad \partial_t k=0, \qquad \mathbf{f}_{\mathrm{bg}}=0, $$ and the derivation of appendix 207 is recovered. ## 213.2 Coarse-Graining in a Variable Background Let $\langle\cdot\rangle$ again denote averaging over a cell large compared to local closure structure and small compared to resolved macroscopic variation. Assume, at the resolved scale, that averaging commutes with differentiation: $$ \langle\partial_t f\rangle=\partial_t\langle f\rangle, \qquad \langle\partial_i f\rangle=\partial_i\langle f\rangle. $$ Then the exact coarse-grained energy balance is still $$ \partial_t\langle u\rangle+\nabla\cdot\langle\mathbf{S}\rangle=0. $$ This remains source-free. Energy itself is still only redistributed. The effective momentum balance is $$ \partial_t\langle\mathbf{g}\rangle \;-\;\nabla\cdot\langle\mathbf{T}\rangle = \langle\mathbf{f}_{\mathrm{bg}}\rangle. $$ So the background enters through momentum exchange, not through failure of energy continuity. ## 213.3 Two Densities: Energy Density and Effective Inertial Density When $k$ varies, it is no longer enough to write $$ \rho=\frac{\langle u\rangle}{k^2} $$ with one constant $k$ pulled out of the cell. The exact effective inertial density is instead $$ \rho:=\langle \beta u\rangle = \left\langle \frac{u}{k^2}\right\rangle. $$ This should be distinguished from the coarse-grained stored-energy density $$ \bar u:=\langle u\rangle. $$ Only $\bar u$ obeys a source-free continuity equation. The quantity $\rho$ is an effective inertial density, and when $k$ varies it does not satisfy a source-free continuity law by itself. Define the coarse-grained momentum density by $$ \rho\mathbf{v}:=\langle\mathbf{g}\rangle = \left\langle\frac{\mathbf{S}}{k^2}\right\rangle. $$ In a slowly varying background, where $k$ changes little across the cell, this reduces to $$ \rho \approx \frac{\bar u}{k(\mathbf{X},t)^2}, \qquad \rho\mathbf{v} \approx \frac{\langle\mathbf{S}\rangle}{k(\mathbf{X},t)^2}, $$ so that $$ \mathbf{v}\approx\frac{\langle\mathbf{S}\rangle}{\langle u\rangle}. $$ Thus the transport velocity remains the ratio of coarse-grained energy flux to coarse-grained energy density, but the effective inertial density now carries the background weighting. ## 213.4 Exact Variable-Background Continuity Equation Differentiate $\rho=\langle\beta u\rangle$: $$ \partial_t\rho = \left\langle \beta\,\partial_t u+u\,\partial_t\beta\right\rangle. $$ Also, $$ \nabla\cdot(\rho\mathbf{v}) = \nabla\cdot\langle\beta\mathbf{S}\rangle = \left\langle \beta\,\nabla\cdot\mathbf{S}+\mathbf{S}\cdot\nabla\beta\right\rangle. $$ Add the two expressions and use $\partial_t u+\nabla\cdot\mathbf{S}=0$: $$ \partial_t\rho+\nabla\cdot(\rho\mathbf{v}) = \left\langle u\,\partial_t\beta+\mathbf{S}\cdot\nabla\beta \right\rangle. $$ Define the exact variable-background source term $$ \sigma_k := \left\langle u\,\partial_t\beta+\mathbf{S}\cdot\nabla\beta \right\rangle. $$ Then $$ \partial_t\rho+\nabla\cdot(\rho\mathbf{v})=\sigma_k. $$ This is the exact balance law for the effective inertial density once $\rho=\langle u/k^2\rangle$ has been adopted. Now write it directly in terms of $k$. Since $$ \beta=k^{-2}, $$ we have $$ \partial_t\beta=-2\beta\,\partial_t\ln k, \qquad \nabla\beta=-2\beta\,\nabla\ln k. $$ So $$ \sigma_k = -2\left\langle \beta\left(u\,\partial_t\ln k+\mathbf{S}\cdot\nabla\ln k\right) \right\rangle. $$ In a slowly varying resolved cell, this becomes $$ \sigma_k \approx -2\rho\left(\partial_t+ \mathbf{v}\cdot\nabla\right)\ln k. $$ Therefore the continuity equation becomes $$ \partial_t\rho+\nabla\cdot(\rho\mathbf{v}) \approx -2\rho\,D_t\ln k, $$ where $$ D_t:=\partial_t+\mathbf{v}\cdot\nabla. $$ This equation says something precise: > when a transported configuration moves into a region where the local > transport scale decreases, the same coarse-grained energy corresponds to a > larger effective inertial density. So the variable-$k$ medium modifies inertia even before any constitutive stress assumption is made. ## 213.5 Exact Coarse-Grained Momentum Equation The coarse-grained momentum balance is $$ \partial_t(\rho\mathbf{v}) \;-\;\nabla\cdot\langle\mathbf{T}\rangle = \mathbf{f}, $$ where $$ \mathbf{f}:=\langle\mathbf{f}_{\mathrm{bg}}\rangle. $$ Define, exactly as before, the residual stress tensor $$ \boldsymbol{\Sigma} := \rho\,\mathbf{v}\otimes\mathbf{v}-\langle\mathbf{T}\rangle. $$ Then $$ \partial_t(\rho\mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) -\nabla\cdot\boldsymbol{\Sigma} = \mathbf{f}. $$ Decompose $$ \boldsymbol{\Sigma}=p\,\mathbf{I}-\boldsymbol{\tau}, $$ with $$ p:=\frac{1}{3}\operatorname{tr}(\boldsymbol{\Sigma}), \qquad \operatorname{tr}(\boldsymbol{\tau})=0. $$ Then the exact variable-background momentum equation is $$ \partial_t(\rho\mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) = -\nabla p+\nabla\cdot\boldsymbol{\tau}+\mathbf{f}. $$ ## 213.6 Convective Form with Variable Background Expand the left-hand side: $$ \partial_t(\rho\mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) = \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) + \mathbf{v}\left(\partial_t\rho+\nabla\cdot(\rho\mathbf{v})\right). $$ Using the exact variable-background continuity equation, $$ \partial_t\rho+\nabla\cdot(\rho\mathbf{v})=\sigma_k, $$ we obtain $$ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p+\nabla\cdot\boldsymbol{\tau}+\mathbf{f}-\mathbf{v}\,\sigma_k. $$ This is the exact convective form in a variable background. In the slowly varying resolved approximation, $$ \rho D_t\mathbf{v} \approx -\nabla p+\nabla\cdot\boldsymbol{\tau} + \mathbf{f} + 2\rho\,\mathbf{v}\,D_t\ln k. $$ So the variable background enters in two distinct ways: - through the explicit momentum-exchange term $\mathbf{f}$, - through the density-conversion term encoded by $\sigma_k$. These two effects should not be conflated. ## 213.7 Static Background Example If the background is static, then $$ \partial_t k=0, $$ and the continuity source reduces to $$ \sigma_k = \langle \mathbf{S}\cdot\nabla\beta\rangle \approx -2\rho\,\mathbf{v}\cdot\nabla\ln k. $$ So $$ \partial_t\rho+\nabla\cdot(\rho\mathbf{v}) \approx -2\rho\,\mathbf{v}\cdot\nabla\ln k. $$ If, in addition, the background momentum exchange is potential-like, write $$ \mathbf{f}=-\rho\,\nabla\Phi_k. $$ Then $$ \rho D_t\mathbf{v} \approx -\nabla p+\nabla\cdot\boldsymbol{\tau} -\rho\,\nabla\Phi_k + 2\rho\,\mathbf{v}\,\mathbf{v}\cdot\nabla\ln k. $$ For the gravity closure of appendix 212, the weak-field metric gave $$ g_{tt}=\frac{k}{c}+O(\eta^2), $$ so the corresponding slow-mode potential is $$ \Phi_k := \frac{c^2}{2}\ln\frac{k}{c}. $$ Since $$ k=c(1-2\eta)+O(\eta^2), $$ we obtain $$ \Phi_k = -c^2\eta+O(\eta^2) = -\frac{GM}{r}+O(\eta^2), $$ and therefore $$ -\nabla\Phi_k = -\frac{GM}{r^2}\,\hat{\mathbf r}+O(\eta^2), $$ which is the Newtonian gravitational acceleration. So the variable-background hydrodynamic form contains the gravity closure of appendix 212 as one special constitutive case. ## 213.8 Euler-like and Navier-Stokes-like Limits in Variable Background The exact equations above reduce to the familiar forms once the unresolved stress is closed. ### Euler-like limit If $$ \boldsymbol{\tau}=0, $$ then $$ \rho D_t\mathbf{v} = -\nabla p+\mathbf{f}-\mathbf{v}\sigma_k. $$ This is the variable-background Euler form. ### Navier-Stokes-like limit If the deviatoric stress is approximated by the Newtonian constitutive form $$ \boldsymbol{\tau} = \eta_v\left(\nabla\mathbf{v}+(\nabla\mathbf{v})^{\mathsf T} -\frac{2}{3}(\nabla\cdot\mathbf{v})\mathbf{I}\right) + \zeta_v(\nabla\cdot\mathbf{v})\mathbf{I}, $$ then $$ \rho D_t\mathbf{v} = -\nabla p+\nabla\cdot\boldsymbol{\tau} + \mathbf{f} -\mathbf{v}\sigma_k. $$ This is the variable-background Navier-Stokes-like form. Compared with the uniform-region case, the new terms are exactly those tied to - background momentum exchange, - variation of the local transport scale. ## 213.9 What Is Exact and What Is Not Up to the introduction of the constitutive form of $\boldsymbol{\tau}$, the derivation is exact once - the variable-background momentum relation $$ \mathbf{g}=\frac{\mathbf{S}}{k^2} $$ is adopted, - the background-exchange term $$ \mathbf{f}_{\mathrm{bg}} $$ is defined as the exact residual momentum exchange with the unresolved background. So the exact conclusions are: - coarse-grained energy density remains conserved, - effective inertial density obeys a balance law with a conversion term when $k$ varies, - the convective momentum equation acquires both an explicit background force term and a density-conversion term. What remains constitutive is: - the local momentum relation $\mathbf{g}=\mathbf{S}/k^2$ for a genuinely variable background, unless it is separately derived from the chosen constitutive closure, as it is in appendix 214 for the symmetric closure, - the exact resolved form of $\mathbf{f}_{\mathrm{bg}}$ for a chosen background closure, - the constitutive closure of the deviatoric stress. ## 213.10 Summary When the local transport scale varies, $$ k=k(\mathbf{r},t), $$ the exact coarse-grained effective inertial density is $$ \rho=\left\langle\frac{u}{k^2}\right\rangle, $$ and the exact momentum density is $$ \rho\mathbf{v} = \left\langle\frac{\mathbf{S}}{k^2}\right\rangle. $$ They satisfy $$ \partial_t\rho+\nabla\cdot(\rho\mathbf{v})=\sigma_k, $$ with $$ \sigma_k = \left\langle u\,\partial_t(k^{-2})+\mathbf{S}\cdot\nabla(k^{-2}) \right\rangle, $$ and $$ \rho D_t\mathbf{v} = -\nabla p+\nabla\cdot\boldsymbol{\tau} + \mathbf{f} -\mathbf{v}\sigma_k. $$ Thus the variable-background hydrodynamic limit is no longer just the uniform-region Euler/Navier-Stokes form with $k(\mathbf r,t)$ inserted by hand. It has a definite new structure: - energy continuity remains source-free, - effective inertia is background-weighted, - background variation creates density-conversion terms, - momentum exchange with the background enters explicitly. This is a consistent variable-background extension of appendix 207 within the matched constitutive relation $\mathbf{g}=\mathbf{S}/k^2$, derived in appendix 214 for the symmetric closure and otherwise still to be fixed by the chosen background structure.