Semigeostrophic Rotating Shallow Water Equations

• Semigeostrophic rotating shallow water equations are a balanced shallow-water model that couples geostrophic momentum approximations with full horizontal advection and continuity constraints.
• The model employs convexity conditions, Monge–Ampère structures, and variational asymptotic methods to ensure stability and well-posedness across both constant and variable Coriolis settings.
• Recent advances integrate entropic regularization and discrete optimal transport techniques to improve computational approaches for capturing ageostrophic corrections and energy dynamics.

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 (Benamou et al., 22 Jul 2025, Cheng et al., 2016). 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

$U_{ag}=U-U_g$0

$U_{ag}=U-U_g$1

The semigeostrophic approximation introduces the geostrophic velocity $U_{ag}=U-U_g$2 through

$U_{ag}=U-U_g$3

and then replaces $U_{ag}=U-U_g$4 by $U_{ag}=U-U_g$5 in the momentum equations, leaving continuity unchanged (Benamou et al., 22 Jul 2025).

In the variable-$U_{ag}=U-U_g$6 semigeostrophic shallow-water system on the flat torus $U_{ag}=U-U_g$7, with $U_{ag}=U-U_g$8, geostrophic velocity $U_{ag}=U-U_g$9, and smooth positive $U$0, the physical-space equations are

$U$1

$U$2

together with

$U$3

In the incompressible semigeostrophic case one has $U$4 instead of the shallow-water mass equation (Cheng et al., 2016).

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 $U$5 and Burger number $U$6, so that height variations satisfy $U$7. The leading-order geostrophic relation is

$U$8

and first-order ageostrophic corrections are obtained from an elliptic balance relation depending on a parameter $U$9 (Dritschel et al., 2017).

2. Geostrophic coordinates, modified pressure, and dual formulations

For constant $f(x)$0, semigeostrophic theory admits a dual formulation in geostrophic coordinates. When $f(x)$1, one introduces the geopotential

$f(x)$2

so that the incompressible semigeostrophic equations become

$f(x)$3

Pushing forward Lebesgue measure by $f(x)$4 yields a density $f(x)$5 satisfying

$f(x)$6

and the convexity condition $f(x)$7 is the hypothesis under which Monge–Ampère theory applies (Cheng et al., 2016).

In the semigeostrophic shallow-water setting on the $f(x)$8-plane, the corresponding modified pressure is

$f(x)$9

and Hoskins’ transform is

$f$0

Strict convexity of $f$1 yields a dual potential $f$2. Writing $f$3 and $f$4, the geostrophic density is

$f$5

and the semigeostrophic dynamics is equivalent to the Lagrangian system

$f$6

or, in Eulerian form on geostrophic space,

$f$7

The same formulation can be written as a Hamiltonian-type flow driven by

$f$8

which couples quadratic transport cost and shallow-water potential energy (Benamou et al., 22 Jul 2025).

For variable $f$9, no global dual coordinates exist. This is the decisive structural distinction between the constant-$f$0 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 (Cheng et al., 2016).

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$1-plane, the Cullen stability principle requires the modified pressure

$f$2

to remain strictly convex for all $f$3 (Benamou et al., 22 Jul 2025).

For variable $f$4, convexity is replaced by positivity of a stability matrix. In the semigeostrophic system one assumes

$f$5

for some $f$6. In the semigeostrophic shallow-water case one also requires $f$7 and

$f$8

Here $f$9 is the Hessian of $U=(u_1,u_2)$0 weighted by $U=(u_1,u_2)$1 (Cheng et al., 2016).

Within the generalized large-scale semigeostrophic family derived by variational asymptotics, the balanced velocity satisfies the elliptic diagnostic relation

$U=(u_1,u_2)$2

As $U=(u_1,u_2)$3, this reduces to geostrophy. For $U=(u_1,u_2)$4, corresponding to Salmon’s $U=(u_1,u_2)$5-model, the balance relation becomes

$U=(u_1,u_2)$6

(Dritschel et al., 2017).

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 $U=(u_1,u_2)$7

Because the dual-space method fails for spatially varying Coriolis parameter, the variable-$U=(u_1,u_2)$8 theory uses a discrete-time Lagrangian flow map. Fix $U=(u_1,u_2)$9 and suppose $h$0 or $h$1 is known. One seeks a measure-preserving diffeomorphism $h$2, periodic in the torus sense, and an updated geopotential $h$3 such that

$h$4

with volume preservation

$h$5

After taking $h$6 and determinants, one obtains a Monge–Ampère-type equation for $h$7,

$h$8

where $h$9 and $g$0 are small commutator matrices of order $g$1 (Cheng et al., 2016).

An implicit-function-theorem argument in Hölder spaces $g$2 gives, for sufficiently small $g$3, a unique solution $g$4 with uniform $g$5-bounds. The limiting continuous result is a short-time existence and uniqueness theorem. If $g$6, $g$7, $g$8, $g$9, and

$f$0

then there exists $f$1 and a unique solution

$f$2

such that

$f$3

The parallel shallow-water result preserves positivity of $f$4 and the normalization $f$5 (Cheng et al., 2016).

The proof strategy consists of constructing time-stepping approximations, obtaining a priori Hölder regularity and Lipschitz-in-time estimates independent of $f$6, passing to the limit by Arzelà–Ascoli, recovering a Lagrangian flow map $f$7 and potential $f$8, differentiating the limiting integral relations to recover the Eulerian equations, and using a Grönwall-type argument in $f$9 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-$U_{ag}=U-U_g$00 system, with absolute vorticity

$U_{ag}=U-U_g$01

the shallow-water potential vorticity is

$U_{ag}=U-U_g$02

and it satisfies the pointwise advection law

$U_{ag}=U-U_g$03

Mass conservation can be written in material form as

$U_{ag}=U-U_g$04

(Cheng et al., 2016).

In the non-dimensional generalized large-scale semigeostrophic models, potential vorticity takes the form

$U_{ag}=U-U_g$05

The standard geostrophic and ageostrophic decomposition is

$U_{ag}=U-U_g$06

with divergence $U_{ag}=U-U_g$07 and ageostrophic vorticity

$U_{ag}=U-U_g$08

A stream-potential representation,

$U_{ag}=U-U_g$09

provides a diagnostic decomposition used in numerical comparisons (Dritschel et al., 2017).

The Lagrangian formulation of the variable-$U_{ag}=U-U_g$10 theory also shows that geostrophic balance is a stationary point of the semigeostrophic energy under volume-preserving Lagrangian displacements (Cheng et al., 2016). 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 $U_{ag}=U-U_g$11 and $U_{ag}=U-U_g$12, the squared Wasserstein cost

$U_{ag}=U-U_g$13

is replaced by the Moreau–Yoshida envelope

$U_{ag}=U-U_g$14

with Gibbs kernel

$U_{ag}=U-U_g$15

The corresponding dual problem yields soft-Kantorovich relations, and the fully discrete approximation replaces both measures by weighted sums of Dirac masses (Benamou et al., 22 Jul 2025).

If

$U_{ag}=U-U_g$16

and $U_{ag}=U-U_g$17, then the discrete entropic cost is minimized over matrices $U_{ag}=U-U_g$18, and the shallow-water energy becomes

$U_{ag}=U-U_g$19

The transport plan is represented in Sinkhorn form,

$U_{ag}=U-U_g$20

with updates

$U_{ag}=U-U_g$21

and, under mild positivity–irreducibility assumptions, these schemes converge at a linear rate in the Hilbert projective metric (Benamou et al., 22 Jul 2025).

Time stepping then couples repeated OT solves to an ODE update in geostrophic space. The barycentric map

$U_{ag}=U-U_g$22

approximates $U_{ag}=U-U_g$23, the discrete geostrophic velocity is

$U_{ag}=U-U_g$24

and one may use Heun’s method or embed the OT solve inside RK4 (Benamou et al., 22 Jul 2025).

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 $U_{ag}=U-U_g$25 against the full rotating shallow-water system over times $U_{ag}=U-U_g$26. The $U_{ag}=U-U_g$27-model, $U_{ag}=U-U_g$28, reproduces the full shallow-water dynamics with

$U_{ag}=U-U_g$29

$U_{ag}=U-U_g$30-error $U_{ag}=U-U_g$31, and $U_{ag}=U-U_g$32-error $U_{ag}=U-U_g$33, while other values $U_{ag}=U-U_g$34 generate significantly larger errors and spurious high-wavenumber power in ageostrophic vorticity, associated with the $U_{ag}=U-U_g$35 term in the $U_{ag}=U-U_g$36-balance relation (Dritschel et al., 2017). A common misconception is that formal $U_{ag}=U-U_g$37 truncation accuracy across the whole one-parameter family is sufficient for long-time balanced fidelity; the numerical comparison shows that only the Salmon $U_{ag}=U-U_g$38-member retains the regularity needed to remain viable on $U_{ag}=U-U_g$39 time scales.