---
title: Semigeostrophic Rotating Shallow Water Equations
url: https://www.emergentmind.com/topics/semigeostrophic-rotating-shallow-water-equations
type: topic
---

# Semigeostrophic Rotating Shallow Water Equations

Semigeostrophic rotating shallow water equations describe a balanced shallow-water regime in which geostrophic momentum balance is coupled to the shallow-water continuity equation while advection is still taken by the full horizontal velocity. On the \(f\)-plane, the semigeostrophic approximation neglects the material acceleration of the ageostrophic part \(U_{ag}=U-U_g\) but retains the material derivative in advection by the full \(U\); with a spatially varying Coriolis parameter \(f(x)\), the two-dimensional semigeostrophic and semigeostrophic shallow-water systems are formulated directly in physical space because the dual-space construction available for constant \(f\) does not exist globally [2507.16415, 1608.05818]. In this setting, convexity or stability, Monge–Ampère structure, Lagrangian flow maps, and potential-vorticity transport are the organizing principles of both analysis and computation.

## 1. Governing equations and semigeostrophic approximation

For the rotating shallow-water equations on the \(f\)-plane, with horizontal velocity \(U=(u_1,u_2)\), layer depth \(h\), gravity \(g\), and constant Coriolis parameter \(f\), the momentum and continuity equations are
\[
\frac{D u_1}{D t}-f\,u_2=-g\,\partial_{x_1}h,\qquad
\frac{D u_2}{D t}+f\,u_1=-g\,\partial_{x_2}h,
\]
\[
\partial_t h+\nabla\!\cdot(h\,U)=0,
\qquad
\frac{D}{D t}=\partial_t+u_1\partial_{x_1}+u_2\partial_{x_2}.
\]
The semigeostrophic approximation introduces the geostrophic velocity \(U_g=(u_{1,g},u_{2,g})\) through
\[
-f\,u_{2,g}=-g\,\partial_{x_1}h,\qquad
f\,u_{1,g}=-g\,\partial_{x_2}h,
\]
and then replaces \(D U/Dt\) by \(D U_g/Dt\) in the momentum equations, leaving continuity unchanged [2507.16415].

In the variable-\(f\) semigeostrophic shallow-water system on the flat torus \(\mathbb T^2=\mathbb R^2/\mathbb Z^2\), with \(u=(u_1,u_2)\), geostrophic velocity \(u_g=(u_{g1},u_{g2})\), and smooth positive \(f=f(x_1,x_2)\), the physical-space equations are
\[
f\,u_{g1}=-\partial_{x_2}h,\qquad
f\,u_{g2}=+\partial_{x_1}h,
\]
\[
D_tu_g-f\,J\,u_g+f\,J\,u=0,\qquad
D_t:=\partial_t+u\!\cdot\!\nabla,\qquad
J=\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\]
together with
\[
\partial_t h+\nabla\!\cdot(h\,u)=0.
\]
In the incompressible semigeostrophic case one has \(\nabla\!\cdot u=0\) instead of the shallow-water mass equation [1608.05818].

A complementary asymptotic derivation arises in the semi-geostrophic scaling of the rotating shallow-water equations. In the variational asymptotic framework, one assumes Rossby number \(\mathrm{Ro}\ll1\) and Burger number \(\mathrm{Bu}=\mathrm{Ro}\), so that height variations satisfy \(h-1=O(1)\). The leading-order geostrophic relation is
\[
u^\perp+\nabla h=0
\qquad\Longleftrightarrow\qquad
u=\nabla^\perp h,
\]
and first-order ageostrophic corrections are obtained from an elliptic balance relation depending on a parameter \(\lambda\) [1705.05579].

## 2. Geostrophic coordinates, modified pressure, and dual formulations

For constant \(f\), semigeostrophic theory admits a dual formulation in geostrophic coordinates. When \(f\equiv1\), one introduces the geopotential
\[
P(x)=p(x)+\tfrac12|x|^2,
\]
so that the incompressible semigeostrophic equations become
\[
D_t(\nabla P)=J\bigl(\nabla P-\mathrm{id}\bigr),\qquad \nabla\!\cdot u=0.
\]
Pushing forward Lebesgue measure by \(\nabla P\) yields a density \(\nu\) satisfying
\[
\partial_t\nu+\nabla\!\cdot(U\,\nu)=0,\qquad
U=J\bigl(\mathrm{id}-\nabla P^*\bigr),
\]
and the convexity condition \(\mathrm D^2P\ge0\) is the hypothesis under which Monge–Ampère theory applies [1608.05818].

In the semigeostrophic shallow-water setting on the \(f\)-plane, the corresponding modified pressure is
\[
P(x,t)=\frac12|x|^2+\frac{g}{f^2}\,h(x,t),
\]
and Hoskins’ transform is
\[
Y=X+\frac{g}{f^2}\,\nabla h(X,t)=\nabla P(X,t).
\]
Strict convexity of \(P\) yields a dual potential \(Q=P^*\). Writing \(\mu=\mathrm dx\) and \(\mu_t=h(\cdot,t)\mu\), the geostrophic density is
\[
\sigma_t=(\nabla P_t)_\#(h_t\mu),
\]
and the semigeostrophic dynamics is equivalent to the Lagrangian system
\[
\dot X_t=U(X_t),\qquad
\dot Y_t=f\,J\bigl(Y_t-X_t\bigr),\qquad
(h_t\mu)=(X_t)_\#(h_0\mu),
\]
or, in Eulerian form on geostrophic space,
\[
\partial_t\sigma_t+\nabla_Y\!\cdot\bigl(\sigma_t\,U_g(\nabla Q_t(Y))\bigr)=0.
\]
The same formulation can be written as a Hamiltonian-type flow driven by
\[
E_\sigma(h)=f^2\,W_2^2(h\mu,\sigma)+\tfrac g2\int h^2,
\]
which couples quadratic transport cost and shallow-water potential energy [2507.16415].

For variable \(f\), no global dual coordinates exist. This is the decisive structural distinction between the constant-\(f\) theory and the variable-Coriolis case, and it is the reason why Cheng, Cullen, and Feldman work in Eulerian and Lagrangian physical space rather than in a global dual Monge–Ampère framework [1608.05818].

## 3. Convexity, stability, and balanced constraints

The convexity requirement is not ancillary; it is the stability hypothesis that makes the semigeostrophic model mathematically and physically coherent. On the \(f\)-plane, the Cullen stability principle requires the modified pressure
\[
P(x,t)=\frac12|x|^2+\frac{g}{f^2}\,h(x,t)
\]
to remain strictly convex for all \(t\in[0,T]\) [2507.16415].

For variable \(f\), convexity is replaced by positivity of a stability matrix. In the semigeostrophic system one assumes
\[
I+f^{-1}D\bigl(f^{-1}Dp_0\bigr)\ge c_0\,I
\quad\text{on }\mathbb T^2,
\]
for some \(c_0>0\). In the semigeostrophic shallow-water case one also requires \(h_0\ge c_1>0\) and
\[
I+f^{-1}D\bigl(f^{-1}Dh_0\bigr)\ge c_0\,I
\quad\text{on }\mathbb T^2.
\]
Here \(D(f^{-1}D\phi)\) is the Hessian of \(\phi\) weighted by \(f^{-2}\) [1608.05818].

Within the generalized large-scale semigeostrophic family derived by variational asymptotics, the balanced velocity satisfies the elliptic diagnostic relation
\[
[1-\epsilon(\lambda+\tfrac12)(h\Delta+2\nabla h\cdot\nabla)]\,u
=
\nabla^\perp\!\left[h-\epsilon\lambda(2h\Delta h+|\nabla h|^2)\right].
\]
As \(\epsilon\to0\), this reduces to geostrophy. For \(\lambda=\tfrac12\), corresponding to Salmon’s \(L_1\)-model, the balance relation becomes
\[
[1-\epsilon(h\Delta+2\nabla h\cdot\nabla)]\,u
=
\nabla^\perp\!\left[h-\tfrac\epsilon2(2h\Delta h+|\nabla h|^2)\right]
\]
[1705.05579].

These conditions formalize a common point that is sometimes obscured in abbreviated presentations: semigeostrophic balance is not merely a geostrophic velocity substitution. It is a constrained regime in which convexity or elliptic stability controls admissibility of the state.

## 4. Lagrangian iteration and short-time well-posedness for variable \(f\)

Because the dual-space method fails for spatially varying Coriolis parameter, the variable-\(f\) theory uses a discrete-time Lagrangian flow map. Fix \(\delta t>0\) and suppose \(p^n\) or \(h^n\) is known. One seeks a measure-preserving diffeomorphism \(F^{n+1}:\mathbb T^2\to\mathbb T^2\), periodic in the torus sense, and an updated geopotential \(p^{n+1}\) such that
\[
x+f^{-1}(x)\,f^{-1}\bigl(F^{n+1}(x)\bigr)\,\nabla p^{n+1}(x)
=
F^{n+1}(x)+\bigl(f^{-2}R_{f\delta t}\nabla p^n\bigr)\bigl(F^{n+1}(x)\bigr),
\]
with volume preservation
\[
\det D F^{n+1}(x)=1.
\]
After taking \(\nabla\) and determinants, one obtains a Monge–Ampère-type equation for \(F^{n+1}\),
\[
\det\Bigl[I+f^{-1}D\bigl(f^{-1}D p^{n+1}\bigr)+A\Bigr]
=
\det\Bigl[\bigl(I+f^{-1}D\bigl(f^{-1}D p^n\bigr)+B\bigr)\circ F^{n+1}\Bigr],
\]
where \(A\) and \(B\) are small commutator matrices of order \(O(\delta t)\) [1608.05818].

An implicit-function-theorem argument in Hölder spaces \(C^{k,\alpha}\) gives, for sufficiently small \(\delta t\), a unique solution \((p^{n+1},F^{n+1})\) with uniform \(C^{k+2,\alpha}\)-bounds. The limiting continuous result is a short-time existence and uniqueness theorem. If \(k\ge2\), \(\alpha\in(0,1)\), \(p_0\in C^{k+2,\alpha}(\mathbb T^2)\), \(\int p_0=0\), and
\[
I+f^{-1}D\bigl(f^{-1}D p_0\bigr)\ge c_0\,I>0,
\]
then there exists \(T>0\) and a unique solution
\[
p\in C\bigl([0,T];C^{k+2,\alpha}(\mathbb T^2)\bigr),\qquad
u\in C\bigl([0,T];C^{k,\alpha}(\mathbb T^2)\bigr),
\]
such that
\[
I+f^{-1}D\bigl(f^{-1}D p(t,\cdot)\bigr)\ge c_0\,I
\quad\text{for all }t\in[0,T].
\]
The parallel shallow-water result preserves positivity of \(h\) and the normalization \(\int_{\mathbb T^2} h=1\) [1608.05818].

The proof strategy consists of constructing time-stepping approximations, obtaining a priori Hölder regularity and Lipschitz-in-time estimates independent of \(\delta t\), passing to the limit by Arzelà–Ascoli, recovering a Lagrangian flow map \(\Phi(t,x)\) and potential \(p(t,x)\), differentiating the limiting integral relations to recover the Eulerian equations, and using a Grönwall-type argument in \(C^{k,\alpha}\) for uniqueness.

## 5. Potential vorticity, energy, and diagnostic structure

Potential-vorticity transport is one of the most persistent invariants across semigeostrophic shallow-water formulations. In the variable-\(f\) system, with absolute vorticity
\[
\zeta=\partial_{x_1}u_2-\partial_{x_2}u_1,
\]
the shallow-water potential vorticity is
\[
q=\frac{\zeta+f(x)}{h},
\]
and it satisfies the pointwise advection law
\[
D_t q=0.
\]
Mass conservation can be written in material form as
\[
D_t h+h\,\nabla\!\cdot u=0
\]
[1608.05818].

In the non-dimensional generalized large-scale semigeostrophic models, potential vorticity takes the form
\[
q=\frac{1+\epsilon\,\nabla^\perp\!\cdot u}{h},
\qquad
\partial_t q+u\cdot\nabla q=0.
\]
The standard geostrophic and ageostrophic decomposition is
\[
u^g=\nabla^\perp h,\qquad
u^{ag}=u-u^g,
\]
with divergence \(\delta=\nabla\!\cdot u\) and ageostrophic vorticity
\[
\gamma=\nabla^\perp\!\cdot u^{ag}=\nabla^\perp\!\cdot u-\Delta h.
\]
A stream-potential representation,
\[
u=\nabla^\perp\psi+\nabla\phi,\qquad
\psi=\Delta^{-1}\gamma+h,\qquad
\phi=\Delta^{-1}\delta,
\]
provides a diagnostic decomposition used in numerical comparisons [1705.05579].

The Lagrangian formulation of the variable-\(f\) theory also shows that geostrophic balance is a stationary point of the semigeostrophic energy under volume-preserving Lagrangian displacements [1608.05818]. This suggests that convexity, PV transport, and variational structure are not separate embellishments but mutually reinforcing aspects of the same balanced dynamics.

## 6. Entropic discretisation and model selection

A recent computational development reformulates semigeostrophic shallow-water dynamics through entropically regularized optimal transport. For probability measures \(\mu\) and \(\nu\), the squared Wasserstein cost
\[
W_2^2(\mu,\nu)
=
\inf_{\pi\in\Pi(\mu,\nu)}\int_{\Omega\times\Omega}|x-y|^2\,\pi(dx,dy)
\]
is replaced by the Moreau–Yoshida envelope
\[
J_\epsilon(\mu,\nu)
=
\inf_{\pi\in\Pi(\mu,\nu)}
\int |x-y|^2\,d\pi(x,y)
+\epsilon\,\mathrm{KL}\!\bigl(\pi\mid\mu\otimes\nu\bigr),
\]
with Gibbs kernel
\[
K_\epsilon(x,y)=\exp\!\bigl(-|x-y|^2/\epsilon\bigr).
\]
The corresponding dual problem yields soft-Kantorovich relations, and the fully discrete approximation replaces both measures by weighted sums of Dirac masses [2507.16415].

If
\[
\mu^N=\sum_{i=1}^N\mu_i\,\delta_{x_i},\qquad
\nu^M=\sum_{j=1}^M\nu_j\,\delta_{y_j},
\]
and \(h\,\mu\approx\sum_i h_i\mu_i\,\delta_{x_i}\), then the discrete entropic cost is minimized over matrices \(P\in U(h,\nu)\), and the shallow-water energy becomes
\[
E_{\nu,\epsilon}(\{h_i\})
=
f^2\,J_\epsilon(\mu^N,\nu^M)
+\frac g2\sum_{i=1}^N h_i^2\,\mu_i.
\]
The transport plan is represented in Sinkhorn form,
\[
P_{ij}=a_i\,K_{ij}\,b_j,
\qquad
K_{ij}=\exp\!\bigl(-|x_i-y_j|^2/\epsilon\bigr),
\]
with updates
\[
a^{(k+1)}=\frac{h\,.\,\mu}{K\,b^{(k)}},\qquad
b^{(k+1)}=\frac{\nu}{K^T a^{(k+1)}},
\]
and, under mild positivity–irreducibility assumptions, these schemes converge at a linear rate in the Hilbert projective metric [2507.16415].

Time stepping then couples repeated OT solves to an ODE update in geostrophic space. The barycentric map
\[
y_i^n
=
x_i+\frac{1}{\nu_i}\sum_j P_{ij}^n\,(y_j-x_i)
\]
approximates \(Y=\nabla P(X)=X+(g/f^2)\nabla h\), the discrete geostrophic velocity is
\[
U_{g,i}^n=f\,J\,(y_i^n-x_i),
\]
and one may use Heun’s method or embed the OT solve inside RK4 [2507.16415].

Model comparison in the semi-geostrophic limit leads to a more selective conclusion. Dritschel, Gottwald, and Oliver compare the generalized large-scale semigeostrophic family for \(\lambda\in\{0,\tfrac12,1,\dots\}\) against the full rotating shallow-water system over times \(t=O(\epsilon^{-1})\). The \(L_1\)-model, \(\lambda=\tfrac12\), reproduces the full shallow-water dynamics with
\[
\|q_{SW}-q_{L1}\|_2=O(\epsilon^2),
\]
\(\delta\)-error \(=O(\epsilon^2)\), and \(\gamma\)-error \(=O(\epsilon^2)\), while other values \(\lambda\neq\tfrac12\) generate significantly larger errors and spurious high-wavenumber power in ageostrophic vorticity, associated with the \(O(\epsilon(\tfrac12-\lambda))\,h\Delta^2 h\) term in the \(\gamma\)-balance relation [1705.05579]. A common misconception is that formal \(O(\epsilon^2)\) truncation accuracy across the whole one-parameter family is sufficient for long-time balanced fidelity; the numerical comparison shows that only the Salmon \(L_1\)-member retains the regularity needed to remain viable on \(O(1/\mathrm{Ro})\) time scales.

Source: https://www.emergentmind.com/topics/semigeostrophic-rotating-shallow-water-equations