Rotating stratified and shallow-water equation families

Let $I\subseteq \mathbb{R}$ be a nonempty open time interval. For $d=2,3$, set $Q_d=[0,2\pi {]}^d$ and define the flat periodic domain ${\mathbb{T}}^d:={\mathbb{R}}^d\mathrm{/}(2\pi \mathbb{Z}{)}^d$. Let ${\lambda }_d^{\ast }$ be the Lebesgue outer set function induced by elementary volume on the algebra ${\mathcal{E}}_d$ of elementary subsets of ${\mathbb{R}}^d$; that is, for $S\subseteq {\mathbb{R}}^d$, ${\lambda }_d^{\ast }(S)=\inf \{{\sum }_{k=0}^{\mathrm{\infty }}{\mu }_0(A_k):A_k\in {\mathcal{E}}_d\text{ for all }k,S\subseteq {\bigcup }_{k=0}^{\mathrm{\infty }}{A}_k\}$, where ${\mu }_0$ is elementary volume. Let $\mathcal{L}({\mathbb{R}}^d)$ be the family of sets $A\subseteq {\mathbb{R}}^d$ satisfying ${\lambda }_d^{\ast }(B)={\lambda }_d^{\ast }(B\cap A)+{\lambda }_d^{\ast }(B\setminus A)$ for every $B\subseteq {\mathbb{R}}^d$, and let ${\lambda }_d$ be the restriction of ${\lambda }_d^{\ast }$ to $\mathcal{L}({\mathbb{R}}^d)$. This is the Carathéodory definition of Lebesgue measure. Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the measure and integration conventions in this item; under this hypothesis $\mathcal{L}({\mathbb{R}}^d)$ is a sigma-algebra, ${\lambda }_d$ is a complete measure, and the box $Q_d$ is measurable with ${\lambda }_d(Q_d)=(2\pi {)}^d$. For a real ${\lambda }_d$-integrable function $F$ on $Q_d$, integrability means measurability and ${\int }_{Q_d}\mathrm{\mid }F\mathrm{\mid }d{\lambda }_d<\mathrm{\infty }$; with $F^{+}=\max (F,0)$ and $F^{-}=\max (-F,0)$, define ${\int }_{Q_d}Fd{\lambda }_d:={\int }_{Q_d}{F}^{+}d{\lambda }_d-{\int }_{Q_d}{F}^{-}d{\lambda }_d$. In formulas, $dx$ means $d{\lambda }_d(x)$ and $dt$ means $d{\lambda }_1(t)$, where ${\lambda }_1$ is Lebesgue measure on $\mathbb{R}$. The expression ${\int }_{{\mathbb{T}}^d}Fdx$ means integration of the periodic lift over $Q_d$; ${\int }_{t_0}^{t_1}{\int }_{Q_d}F(t,x)dxdt$ means the iterated Lebesgue integral with the spatial integral first whenever it exists. A field on ${\mathbb{T}}^d$ is represented by its $2\pi$-periodic lift to the nonempty open spatial domain ${\mathbb{R}}^d$. All fields below are classical: their displayed time and spatial derivatives are continuous, and the velocity and buoyancy fields have continuous second spatial derivatives. Each displayed equation is a pointwise equality.

For a scalar field $q$, set $\mathrm{\nabla }q=({\mathrm{\partial }}_{x_1}q,\dots ,{\mathrm{\partial }}_{x_d}q)$ and $\mathrm{\Delta }q={\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}^{2}q$. For a vector field $w=(w_1,\dots ,w_d)$, set $\mathrm{\nabla }\cdot w={\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}{w}_j$, $\mathrm{\Delta }w=(\mathrm{\Delta }{w}_1,\dots ,\mathrm{\Delta }{w}_d)$, $\mathrm{\mid }w{\mathrm{\mid }}^2={\sum }_{j=1}^d{w}_j^2$, and $\mathrm{\mid }\mathrm{\nabla }w{\mathrm{\mid }}^2={\sum }_{i,j=1}^d({\mathrm{\partial }}_{x_j}{w}_i{)}^2$. For a matrix field $A$, its divergence is rowwise: $(\mathrm{\nabla }\cdot A{)}_i={\sum }_{j=1}^d{\mathrm{\partial }}_{x_j}{A}_{ij}$. In three dimensions $e_3\times u=(-u_2,u_1,0)$, and in two dimensions $Jv=(-v_2,v_1)$.

On ${\mathbb{T}}^3$, let $u:I\times {\mathbb{T}}^3\to {\mathbb{R}}^3$ be velocity, $b:I\times {\mathbb{T}}^3\to \mathbb{R}$ be a buoyancy perturbation, and $\pi :I\times {\mathbb{T}}^3\to \mathbb{R}$ be specific pressure. Let $e_3=(0,0,1)$, and prescribe constant parameters $f\in \mathbb{R}$, $N>0$, and $\nu ,\kappa \ge 0$. The rotating, linear, constant-stratification Boussinesq equation family is

$u_t+f{e}_3\times u+\mathrm{\nabla }\pi =b{e}_3+\nu \mathrm{\Delta }u,\mathrm{\nabla }\cdot u=0,b_t+N^2{u}_3=\kappa \mathrm{\Delta }b\mathrm{.}$

Here $\mathrm{\Delta }$ acts componentwise, $u_3$ is the vertical velocity component, $f$ is the Coriolis coefficient, $N$ is the constant buoyancy-frequency parameter, and $\nu$ and $\kappa$ are the prescribed momentum and buoyancy diffusion coefficients. The inviscid branch is the choice $\nu =\kappa =0$. In this mathematical family, $b$ is only the scalar acceleration variable appearing in the vertical forcing term $b{e}_3$ and in its evolution equation; no density field or identification such as $b=-g\delta \rho \mathrm{/}{\rho }_0$ is part of this definition. The positive parameter $N$ has units ${\mathrm{s}}^{-1}$ and its square is the prescribed coefficient coupling $u_3$ to $b$; no background buoyancy profile $b_0(z)$ or identification $N^2={\mathrm{\partial }}_z{b}_0$ is included. Such density-anomaly and background-profile relations would be additional physical modeling assumptions. In the formal Boussinesq interpretation, a constant reference density is used in inertia and the divergence constraint while density-anomaly effects are retained only through the vertical buoyancy acceleration; small fractional density variation is an applicability assumption, not a theorem giving a compressible limit.

For its rotating-frame interpretation, let $\mathbf{E}$ be an oriented Euclidean three-space with a fixed origin and fixed upward unit vector $\widehat{\mathbf{z}}$. A rotating basis on $I$ is a triple of $C^1$ maps ${\mathbf{e}}_a:I\to \mathbf{E}$ ($a=1,2,3$) such that $⟨{\mathbf{e}}_a(t),{\mathbf{e}}_b(t)⟩={\delta }_{ab}$, the ordered basis is positively oriented, ${\mathbf{e}}_3(t)=\widehat{\mathbf{z}}$, and ${\dot{\mathbf{e}}}_a(t)=\mathbf{\Omega }\times {\mathbf{e}}_a(t)$ for $a=1,2$, with constant $\mathbf{\Omega }=\mathrm{\Omega }\widehat{\mathbf{z}}$. This differential equation fixes the sign convention for the signed angular-velocity scalar $\mathrm{\Omega }$; for this interpretation, spatial coordinates $x$ and vector components are relative to this time-indexed basis, and $u_t$ differentiates the components at fixed rotating coordinates. The component triple $u$ represents the vector ${\sum }_{a=1}^3{u}_a{\mathbf{e}}_a(t)$ in $\mathbf{E}$. Under this convention the Coriolis acceleration is $-2\mathbf{\Omega }\times u$ on the force side, hence $+f{e}_3\times u$ on the left with $f=2\mathrm{\Omega }$. Let $\mathrm{\Phi }:I\times {\mathbb{R}}^3\to \mathbb{R}$ be a gravitational potential per unit mass in the rotating coordinates, with continuous first spatial derivatives; $x\in {\mathbb{R}}^3$ denotes the coordinate vector relative to the fixed origin in the basis ${\mathbf{e}}_a(t)$. Define the effective potential ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{f}\mathrm{f}}:I\times {\mathbb{R}}^3\to \mathbb{R}$ by ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{f}\mathrm{f}}(t,x):=\mathrm{\Phi }(t,x)-\frac{1}{2}\mathrm{\mid }\mathbf{\Omega }\times x{\mathrm{\mid }}^2$, which incorporates the centrifugal contribution. These rotating-frame and effective-potential choices are assumptions; the inertial-frame balance is distinct. The periodic domain is a mathematical idealization, not a claim that the physical fluid occupies a compact torus.

Independently, on ${\mathbb{T}}^2$, let $h:I\times {\mathbb{T}}^2\to (0,\mathrm{\infty })$ be total layer depth and $v:I\times {\mathbb{T}}^2\to {\mathbb{R}}^2$ be horizontal velocity. Let $g>0$ and $f\in \mathbb{R}$ be prescribed constants, define $Jv:=(-v_2,v_1)$, and take a fixed flat bottom. The inviscid one-layer shallow-water equation family is

$h_t+\mathrm{\nabla }\cdot (hv)=0,(hv{)}_t+\mathrm{\nabla }\cdot (hv\otimes v)+\mathrm{\nabla }\text{\hspace{0.17em}⁣}\left(\frac{g{h}^2}{2}\right)+fhJv=0.$

The tensor divergence is rowwise, $(\mathrm{\nabla }\cdot (hv\otimes v){)}_i={\sum }_{j=1}^2{\mathrm{\partial }}_{x_j}(h{v}_i{v}_j)$. This family has one layer, positive depth, a flat bottom, and no bottom friction, wetting/drying, or interlayer coupling. These are mathematical equation systems, not assertions that a measured fluid realizes them. With SI interpretation, $t$ is in seconds, spatial coordinates in metres, $u$ and $v$ in metres per second, $b$ in metres per second squared, $\pi$, $\mathrm{\Phi }$, and ${\mathrm{\Phi }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in square metres per square second, $f$, $N$, and $\mathrm{\Omega }$ in inverse seconds, $\nu$ and $\kappa$ in square metres per second, $h$ in metres, and $g$ in metres per square second. The displayed coefficients are prescribed within these formulations; no empirical calibration is asserted. If a spatially integrated energy for either system is defined per unit constant reference density, multiplying that normalized energy by the specified density in kilograms per cubic metre gives joules.

Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, assumed for the measure and integration conventions: The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Lebesgue outer measure as the infimum of elementary-volume costs over countable elementary covers: Lebesgue outer measure on ${\mathbb{R}}^n$.

Lebesgue measure on ${\mathbb{R}}^d$, including the real-line measure ${\lambda }_1$: Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$.

Countable additivity and completeness of Lebesgue measure under ${\mathrm{A}\mathrm{C}}_{\omega }$: Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume.

Measurability and measure of a closed box under ${\mathrm{A}\mathrm{C}}_{\omega }$: A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included.

Integrability and the signed Lebesgue integral of real functions: Integrable real and complex functions, and their integrals.

Formal-PDE supplier Formal parameterized fluid systems and observables: it defines $I$ as a nonempty open time interval and $\mathrm{\Omega }$ as a nonempty open subset of ${\mathbb{R}}^3$, requires pointwise classical equations with velocity $C^1$ in time and $C^2$ in space and pressure $C^1$ in space, and requires any further regularity used by a proof. Its Cartesian gradient, divergence, Laplacian, componentwise vector-Laplacian, row-component Jacobian, and Euclidean conventions match the conventions stated locally above. The periodic lifts use $\mathrm{\Omega }={\mathbb{R}}^3$, so the stratified branch meets its spatial-domain hypothesis. It does not define the two-dimensional shallow-water equations, which this item specifies locally.

Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, University of Exeter full lecture notes, §§1.6.2–1.6.3 (buoyancy frequency and Boussinesq energetics) and §2.4.2 (shallow-water energy): full lecture notes, PDF.

Darryl D. Holm, Ideal shallow water dynamics in a rotating frame, Imperial College London MPE CDT PDE full lecture notes, §§1.1 and 1.4 (rotating equations and their energy law): full lecture notes, PDF.

Paola Rizzoli, MIT OpenCourseWare 12.802, Lecture 12, Shallow water equations with rotation — Poincaré waves, pp. 1–4 (derivation of the rotating one-layer system): full lecture notes, PDF.