Primitive Equations in Geophysics

• Primitive Equations are a set of nonlinear PDEs derived under the hydrostatic approximation that model oceanic and atmospheric dynamics using horizontal momentum balance and diagnosed vertical motion.
• They are rigorously obtained as asymptotic limits of the Navier–Stokes, Boussinesq, and compressible models, with explicit convergence rates and well-posedness results established in various functional spaces.
• The equations support diverse reformulations—incorporating anisotropic viscosity, stochastic forcing, and symmetry transformations—which provide deep insights into long-term statistical behavior and analytical structure.

Primitive equations are a system of nonlinear partial differential equations that arise under the hydrostatic approximation in large-scale oceanic and atmospheric dynamics. In the height-variable formulation they combine horizontal momentum, hydrostatic balance, and incompressibility; in the pressure-coordinate formulation they combine horizontal momentum, hydrostatic balance, mass continuity, and thermodynamic evolution. Across the literature summarized here, they appear as the hydrostatic limit of anisotropic Navier–Stokes equations, of scaled Boussinesq equations with rotation, and as the incompressible limit of compressible primitive equations. They also admit geometric, stochastic, and symmetry-based reformulations, and they occupy a central position in mathematical geophysics because global strong well-posedness, hydrostatic approximation, anisotropic dissipation, and long-time statistical behavior can all be studied within the same framework (Li et al., 2017, Pu et al., 2021, Lemarié, 2024).

1. Canonical formulations

In a Cartesian horizontal strip $M=\{(x_1,x_2,z)\,;\,0\le z\le H\}$ with no-flux boundary $u_3|_{z=0,H}=0$, the inviscid rotating stratified primitive equations read

$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$

where $\u=(u_1,u_2,u_3)^T$, $\u_h=(u_1,u_2,0)^T$, $\u^\perp=(-u_2,u_1,0)^T$, $f(x)$ is the Coriolis frequency, $\theta$ is buoyancy (or potential temperature), and $p$ is pressure (Badin et al., 2018).

A viscous hydrostatic formulation on $\Omega_i\times[0,\infty)$ is

$u_3|_{z=0,H}=0$0

with periodic or strip boundary/parity conditions depending on the domain (Lemarié, 2024). A rotating, anisotropic formulation derived from the incompressible Navier–Stokes equations in a rotating frame with Coriolis parameter $u_3|_{z=0,H}=0$1 is

$u_3|_{z=0,H}=0$2

with $u_3|_{z=0,H}=0$3 (Saal, 2018).

In the pressure-coordinate form, with $u_3|_{z=0,H}=0$4 as the vertical coordinate and $u_3|_{z=0,H}=0$5, the frictionless primitive equations include

$u_3|_{z=0,H}=0$6

$u_3|_{z=0,H}=0$7

$u_3|_{z=0,H}=0$8

together with mass continuity (Cardoso-Bihlo et al., 2015).

These formulations share the same structural core: horizontal momentum is prognostic, vertical motion is diagnosed from incompressibility, and the pressure satisfies a hydrostatic or hydrostatically reduced balance. A plausible implication is that many apparently different primitive-equation models differ less by their principal balance laws than by their choices of viscosity, thermodynamic coupling, coordinates, or stochastic closure.

2. Hydrostatic approximation and asymptotic derivations

A standard route to the primitive equations is the small-aspect-ratio limit of incompressible Navier–Stokes equations. On a thin domain of aspect ratio $u_3|_{z=0,H}=0$9, after appropriate rescaling one obtains

$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$0

$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$1

$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$2

Formally letting $\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$3 forces $\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$4 and yields the hydrostatic primitive equations (Furukawa et al., 2018). In the periodic setting, Li and Titi proved that the scaled Navier–Stokes equations converge strongly to the primitive equations, globally and uniformly in time, and the convergence rate is of the same order as the aspect ratio parameter (Li et al., 2017). In a maximal-$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$5-$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$6-regularity setting, Furukawa, Giga, Hieber, Hussein, Kashiwabara, and Wrona proved

$\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$7

with initial data in $\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$8 under stated restrictions on $\begin{aligned} &\partial_t \u_h + \nabla_{\u}\u_h + f(x)\,\u^\perp + \theta\,\e_z + \nabla p = 0,\ &\partial_t \theta + \nabla_{\u}\theta = 0,\ &\nabla\cdot\u = 0, \end{aligned}$9 (Furukawa et al., 2018).

A second route starts from the scaled Boussinesq equations with rotation. On $\u=(u_1,u_2,u_3)^T$0,

$\u=(u_1,u_2,u_3)^T$1

$\u=(u_1,u_2,u_3)^T$2

Formally letting $\u=(u_1,u_2,u_3)^T$3 yields the full primitive equations

$\u=(u_1,u_2,u_3)^T$4

with temperature evolution and incompressibility (Pu et al., 2021). Pu and Zhou proved that the scaled Boussinesq equations with rotation converge to the full primitive equations in a strong sense, globally in time, with the convergence rate $\u=(u_1,u_2,u_3)^T$5 (Pu et al., 2021).

A third route is the zero Mach number limit of the compressible primitive equations. Liu and Titi identify the primitive equations with the incompressibility condition as the limiting equations, rigorously justify the convergence with well-prepared initial data, and show that the convergence rate is of order $\u=(u_1,u_2,u_3)^T$6 as the Mach number tends to zero (Liu et al., 2019). Their formulation includes

$\u=(u_1,u_2,u_3)^T$7

$\u=(u_1,u_2,u_3)^T$8

and in the limit recovers the incompressible primitive equations (Liu et al., 2019).

These derivations clarify a common misconception. The primitive equations are not merely an ad hoc truncation of Navier–Stokes dynamics; in the cited settings they are rigorously justified as asymptotic limits, with explicit norms and convergence rates (Li et al., 2017, Furukawa et al., 2018, Pu et al., 2021, Liu et al., 2019).

3. Well-posedness, critical regularity, and analyticity

The primitive equations admit several global well-posedness theories, depending on geometry, viscosity, and function space. On cylindrical domains $\u=(u_1,u_2,u_3)^T$9, Giga, Gries, Hieber, Hussein, and Kashiwabara proved that the $\u_h=(u_1,u_2,0)^T$0 primitive equations admit a unique, global strong solution for initial data lying in the critical solonoidal Besov space $\u_h=(u_1,u_2,0)^T$1 for $\u_h=(u_1,u_2,0)^T$2 with $\u_h=(u_1,u_2,0)^T$3, and that this solution regularize instantaneously and becomes even real analytic for $\u_h=(u_1,u_2,0)^T$4 (Giga et al., 2017). Their framework uses the hydrostatic Stokes operator $\u_h=(u_1,u_2,0)^T$5, maximal $\u_h=(u_1,u_2,0)^T$6-regularity, and the critical trace space

$\u_h=(u_1,u_2,0)^T$7

which is invariant under the parabolic scaling $\u_h=(u_1,u_2,0)^T$8 (Giga et al., 2017).

In critical Besov spaces $\u_h=(u_1,u_2,0)^T$9, Lemarié proved an existence theorem for global solutions on a suitable Besov space for the primitive equations on $\u^\perp=(-u_2,u_1,0)^T$0 with periodic boundary conditions and on the strip $\u^\perp=(-u_2,u_1,0)^T$1 with periodic vertical coordinate (Lemarié, 2024). Under small data,

$\u^\perp=(-u_2,u_1,0)^T$2

and

$\u^\perp=(-u_2,u_1,0)^T$3

for all $\u^\perp=(-u_2,u_1,0)^T$4 (Lemarié, 2024).

In an infinite-layer domain $\u^\perp=(-u_2,u_1,0)^T$5, Hussein, Saal, and Sawada considered primitive equations with linearly growing initial data

$\u^\perp=(-u_2,u_1,0)^T$6

where $\u^\perp=(-u_2,u_1,0)^T$7 is constant with $\u^\perp=(-u_2,u_1,0)^T$8. After the change of variables $\u^\perp=(-u_2,u_1,0)^T$9, the linear part becomes an Ornstein–Uhlenbeck type operator, and local existence and uniqueness of mild solutions are proved by an adapted Fujita–Kato scheme in certain Sobolev spaces (Hussein et al., 2017). This setting models uniform rotation or horizontal straining and extends the class of admissible large-scale flows beyond spatially decaying data.

For physical boundary conditions, Evans and Gastler showed that the simplified $f(x)$0-dimensional primitive equations with constant forcing have a bounded absorbing ball in the $f(x)$1-norm and that a solution to the unforced equations has its $f(x)$2-norm decay to $f(x)$3 (Evans et al., 2011). Their results are formulated on a bounded cylindrical domain with

$f(x)$4

and they use $f(x)$5-energy estimates, Poincaré’s inequality, and a uniform Grönwall argument (Evans et al., 2011).

Taken together, these results show that primitive-equation well-posedness is compatible with critical Besov regularity, maximal regularity, real analyticity for $f(x)$6, linearly growing background flows, and physical boundary conditions. This suggests that the hydrostatic structure, rather than weakening analysis, often strengthens it relative to the $f(x)$7 Navier–Stokes setting.

4. Anisotropic viscosity, $f(x)$8-models, and geometric structure

A distinctive feature of primitive-equation analysis is anisotropy. Saal showed that one does need less than horizontal viscosity to obtain a well-posedness result in Sobolev spaces (Saal, 2018). For full horizontal viscosity with $f(x)$9,

$\theta$0

and for $\theta$1, $\theta$2, one has a unique strong solution with

$\theta$3

for all $\theta$4, and when $\theta$5 the solution is global in time (Saal, 2018). In the pure-horizontal-viscosity case no boundary conditions on $\theta$6 at $\theta$7 are required, because $\theta$8 acts like transport by a velocity $\theta$9 which vanishes at $p$0 (Saal, 2018). For half horizontal viscosity, local well-posedness is obtained under a Rayleigh condition or related structural assumptions (Saal, 2018).

A separate anisotropic limit arises when the vertical viscosity in the incompressible $p$1 Navier–Stokes equations is of order $p$2 with $p$3. Li, Titi, and Yuan proved that the limiting system is the primitive equations with only horizontal viscosity as $p$4 tends to zero, and that the convergence rate is of order $p$5, where $p$6 (Li et al., 2021). This differs from the case $p$7, where the limiting system is the primitive equations with full viscosities and the convergence is globally in time with rate $p$8 (Li et al., 2021).

Badin, Oliver, and Vasylkevych derived a horizontally-isotropic Lagrangian-averaged primitive-equations $p$9-model from the geometric generalized Lagrangian mean and an Euler–Poincaré variational framework (Badin et al., 2018). Under the generalized Taylor hypothesis

$\Omega_i\times[0,\infty)$0

and horizontal isotropy of fluctuations

$\Omega_i\times[0,\infty)$1

the reduced averaged Lagrangian becomes

$\Omega_i\times[0,\infty)$2

with horizontal $\Omega_i\times[0,\infty)$3-regularization (Badin et al., 2018). The resulting horizontally-isotropic Lagrangian-averaged primitive equations read

$\Omega_i\times[0,\infty)$4

where

$\Omega_i\times[0,\infty)$5

Because the averaged Lagrangian is invariant under the same particle-relabeling as the parent primitive equations, one obtains conservation of

$\Omega_i\times[0,\infty)$6

and material potential vorticity

$\Omega_i\times[0,\infty)$7

with $\Omega_i\times[0,\infty)$8 (Badin et al., 2018).

The comparison with standard PE-$\Omega_i\times[0,\infty)$9 is precise. The classical primitive-equations-$u_3|_{z=0,H}=0$00 model uses the full $u_3|_{z=0,H}=0$01 Laplacian $u_3|_{z=0,H}=0$02 instead of the horizontal $u_3|_{z=0,H}=0$03 in the definition of $u_3|_{z=0,H}=0$04, and it does not modify the Coriolis term (Badin et al., 2018). By contrast, the HILAPE model requires no “extra” boundary conditions for $u_3|_{z=0,H}=0$05, preserves the same geometric and Hamiltonian structure, but regularizes only in horizontal directions (Badin et al., 2018).

5. Stochastic primitive equations and long-time statistical behavior

The stochastic theory of primitive equations includes additive noise, multiplicative noise, transport noise, Gaussian invariant measures, and moderate deviations. In two dimensions, Zhang and Zhou established a central limit theorem and a moderate deviation principle for $u_3|_{z=0,H}=0$06 stochastic primitive equations with multiplicative noise (Zhang et al., 2017). After eliminating $u_3|_{z=0,H}=0$07 and $u_3|_{z=0,H}=0$08, their stochastic evolution system for $u_3|_{z=0,H}=0$09 on $u_3|_{z=0,H}=0$10 is

$u_3|_{z=0,H}=0$11

with

$u_3|_{z=0,H}=0$12

They proved global well-posedness in $u_3|_{z=0,H}=0$13, a CLT for

$u_3|_{z=0,H}=0$14

and a moderate deviation principle for

$u_3|_{z=0,H}=0$15

with speed $u_3|_{z=0,H}=0$16 (Zhang et al., 2017).

For $u_3|_{z=0,H}=0$17 stochastic primitive equations driven by linear multiplicative noise under non-periodic boundary conditions, Zhang and Zhou proved the existence of a random attractor and then the existence of invariant measure (Zhang et al., 2018). Their Stratonovich system includes

$u_3|_{z=0,H}=0$18

$u_3|_{z=0,H}=0$19

$u_3|_{z=0,H}=0$20

Because Sobolev imbedding does not directly yield tightness in this non-periodic setting, the proof proceeds through compactness and regularity of the solution operator, the construction of a random attractor, and then an invariant measure (Zhang et al., 2018).

Grotto and Pappalettera introduced a Gaussian measure formally preserved by the $u_3|_{z=0,H}=0$21-dimensional Primitive Equations driven by additive Gaussian noise (Grotto et al., 2020). In hyperviscous form,

$u_3|_{z=0,H}=0$22

Writing $u_3|_{z=0,H}=0$23, they reduce the system to the scalar SPDE

$u_3|_{z=0,H}=0$24

The centered Gaussian measure $u_3|_{z=0,H}=0$25 with covariance operator $u_3|_{z=0,H}=0$26 is formally invariant; in the critical regime $u_3|_{z=0,H}=0$27, $u_3|_{z=0,H}=0$28, so $u_3|_{z=0,H}=0$29 is space white noise at the level of $u_3|_{z=0,H}=0$30 (Grotto et al., 2020). Because the nonlinearity is singular under $u_3|_{z=0,H}=0$31, existence and uniqueness are proved only in the hyperviscous regime: if $u_3|_{z=0,H}=0$32, there exists a stationary martingale solution $u_3|_{z=0,H}=0$33 with marginals $u_3|_{z=0,H}=0$34; if $u_3|_{z=0,H}=0$35, this solution is pathwise unique in $u_3|_{z=0,H}=0$36 (Grotto et al., 2020).

These results establish that primitive-equation stochasticity is not limited to perturbative forcing. It also supports large-deviation asymptotics, invariant measures, random attractors, and singular equilibrium theories.

6. Symmetry, non-hydrostatic relaxation, and related generalizations

Dos Santos Cardoso, Bihlo, and Popovych computed Lie symmetries of the primitive equations with zero external heating rate and studied the structure of the maximal Lie invariance algebra, which is infinite-dimensional (Cardoso-Bihlo et al., 2015). A key result is that the primitive equations with constant Coriolis parameter $u_3|_{z=0,H}=0$37 can be mapped pointwise to those with $u_3|_{z=0,H}=0$38 by the invertible change of variables

$u_3|_{z=0,H}=0$39

$u_3|_{z=0,H}=0$40

$u_3|_{z=0,H}=0$41

This mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame and to obtain exact solutions for the rotating case from exact solutions for the nonrotating equations (Cardoso-Bihlo et al., 2015). The same work computes the complete point symmetry group by the algebraic method (Cardoso-Bihlo et al., 2015).

Recent work also studies models that sit between incompressible $u_3|_{z=0,H}=0$42 LU Navier–Stokes equations and standard LU primitive equations by relaxing the classical hydrostatic balance hypothesis. Debussche, Mémin, and Moneyron consider a non-hydrostatic stochastic oceanic primitive-equations model on $u_3|_{z=0,H}=0$43, with weak low-pass filtered hydrostatic hypothesis

$u_3|_{z=0,H}=0$44

$u_3|_{z=0,H}=0$45

and stochastic transport operator

$u_3|_{z=0,H}=0$46

Under rigid-lid type boundary conditions and when the horizontal component of noise is independent of $u_3|_{z=0,H}=0$47, they prove well-posedness of a specific stochastic interpretation of the LU primitive equations and show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise (Debussche et al., 2024).

The 2025 continuation studies how weakening the classical hydrostatic balance hypothesis impacts the well-posedness of the stochastic LU primitive equations and presents two eddy-(hyper)viscosity-based models (Debussche et al., 20 Feb 2025). Under the weak hydrostatic hypothesis,

$u_3|_{z=0,H}=0$48

and with low-pass regularization $u_3|_{z=0,H}=0$49, one obtains global martingale solutions in $u_3|_{z=0,H}=0$50 and global pathwise solutions in the quasi-barotropic case when the horizontal noise is purely barotropic (Debussche et al., 20 Feb 2025).

A related stochastic generalization with non-isothermal turbulent pressure introduces transport noise and a temperature-dependent turbulent pressure term. On $u_3|_{z=0,H}=0$51, the Itô-form stochastic primitive equations include

$u_3|_{z=0,H}=0$52

$u_3|_{z=0,H}=0$53

$u_3|_{z=0,H}=0$54

Global well-posedness in $u_3|_{z=0,H}=0$55 is proved in both the Itô and Stratonovich formulations (Agresti et al., 2022).

These developments show that the hydrostatic balance can be treated as a limit, as a symmetry-compatible reduction, or as a hypothesis to be relaxed in controlled ways. This suggests that the primitive equations are best understood not as a single PDE system, but as a mathematically coherent family of hydrostatic and weakly non-hydrostatic models connected by asymptotic, geometric, and stochastic structure.