Dimensionless Navier–Stokes Equations

• Dimensionless Navier–Stokes equations are a rescaled framework capturing key fluid dynamics under stratification using non-dimensional parameters.
• The formulation isolates critical groups like Reynolds, Péclet, and Froude numbers, emphasizing the low-Froude-number regime with strong stratification.
• The reduction to two-dimensional stratified dynamics underscores hydrostatic balance and aids in simplified modeling of geophysical flows.

The dimensionless Navier–Stokes equations provide a systematic framework for analyzing the incompressible, density-dependent flow of fluids subject to gravitational forces, under the influence of viscosity and scalar diffusion. By rescaling the canonical Navier–Stokes–Boussinesq system with physically relevant characteristic quantities, all parameters are rendered dimensionless, isolating the role of key non-dimensional groups: the Reynolds number, Péclet number, and Froude number. The particular regime where the Froude number is small (strong stratification) exhibits singular asymptotic behavior, with major consequences for the underlying flow structure and its mathematical treatment, as established in rigorous work for the low-Froude-number limit (Scrobogna, 2017).

1. Formulation of the Dimensional Boussinesq System

The starting point is the incompressible, density-dependent Navier–Stokes system with uniform gravity (the Boussinesq approximation). The variables are:

• $u=(u_1,u_2,u_3)$: velocity field,
• $\rho(x,t)=\rho_0+\bar\rho(x_3)+\rho'(x,t)$: total density, with $\bar\rho(x_3)$ a stable stratification,
• $p$: kinematic pressure,
• $\nu, \kappa$: kinematic viscosity and diffusivity, both constant,
• $g$: gravitational acceleration.

The Brunt–Väisälä frequency $N^2=-\frac{g}{\rho_0}\frac{d\bar\rho}{dx_3}$ quantifies the stratification. The governing equations (see eqs. (2.7) in (Scrobogna, 2017)) are: $$\begin{align*} &\partial_t u + u\cdot\nabla u + \frac{1}{\rho_0} \nabla p - \nu \Delta u = -\frac{g}{\rho_0} \rho' e_3, \ &\partial_t \rho' + u\cdot\nabla \rho' + N^2 u_3 - \kappa \Delta \rho' = 0, \ &\nabla \cdot u = 0, \end{align*}$$ where $e_3 = (0,0,1)$ and $N^2 u_3$ introduces the stabilizing effect of stratification.

2. Nondimensionalization: Scales and Dimensionless Groups

Characteristic scales are selected as follows:

• $L$: length scale,
• $U$: velocity scale,
• $T=L/U$: advection time scale,
• $T_N=1/N$: stratification (Brunt–Väisälä) time scale.

The dimensionless numbers are then defined as:

• Froude number: $\mathrm{Fr}=T_N/T=U/(N L)$, comparing advection to stratification,
• Reynolds number: $\mathrm{Re} = UL / \nu$, ratio of advective to viscous effects,
• Péclet number: $\mathrm{Pe} = UL / \kappa$, ratio of advective to diffusive effects for scalars.

In the target asymptotic regime, the Froude number is set as $\varepsilon$ with $\varepsilon \ll 1$ to indicate strong stratification.

3. Nondimensional Variables and Parameterization

The rescaling (with $x'=x/L$, $t'=t/T$, $u'=u/U$, $p'=p / (\rho_0 U^2)$, and with primes then suppressed) yields:

• $x = L x'$, $t = T t'$, $u = U u'$, $p = \rho_0 U^2 p'$, $\rho' = \rho_{\text{scale}} \rho''$,
• with $$\rho_{\text{scale}}=(\rho_0 U^2)/(g L)=\rho_0 \mathrm{Fr}^2$$ (since $gL=N^2L^2=(U \mathrm{Fr})^2$).

Dimensionless coefficients become $\nu^*=1/\mathrm{Re}$, $\kappa^*=1/\mathrm{Pe}$.

4. Full Dimensionless Navier–Stokes–Boussinesq System

Expressed in compact form with $U^\varepsilon=(u^\varepsilon, p^\varepsilon)^T$, a skew-symmetric coupling operator $A$, and $$D=\text{diag}(\nu \Delta, \nu \Delta, \nu \Delta, \kappa \Delta)$$: $$\partial_t U^\varepsilon + u^\varepsilon \cdot \nabla U^\varepsilon - D U^\varepsilon + \frac{1}{\varepsilon} A U^\varepsilon = 0, \qquad \nabla \cdot u^\varepsilon = 0.$$ Expanding, this yields the explicit system: $$\begin{align*} &\partial_t u^\varepsilon + u^\varepsilon \cdot \nabla u^\varepsilon - \frac{1}{\mathrm{Re}} \Delta u^\varepsilon + \frac{1}{\varepsilon} \nabla p^\varepsilon = -\frac{1}{\varepsilon} \rho^\varepsilon e_3,\ &\nabla \cdot u^\varepsilon = 0,\ &\partial_t \rho^\varepsilon + u^\varepsilon \cdot \nabla \rho^\varepsilon - \frac{1}{\mathrm{Pe}} \Delta \rho^\varepsilon + \frac{1}{\varepsilon} u_3^\varepsilon = 0. \end{align*}$$ Here, $\Delta \equiv \Delta_x$ and all variables are nondimensional.

5. The Low Froude Number Limit and Hydrostatic Balance

In the singular limit $\varepsilon \to 0$, the $O(1/\varepsilon)$ fast terms enforce:

• Hydrostatic balance in the vertical: $\nabla p^\varepsilon \approx -\rho^\varepsilon e_3$,
• Suppression of vertical velocity: $u_3^\varepsilon \approx 0$.

This implies that, at leading order, the vertical velocity component vanishes and the pressure becomes slaved to the density fluctuation, enforcing hydrostatic equilibrium. The flow dynamics decouple between horizontal and vertical directions.

6. Limit System: Two-Dimensional Stratified Dynamics

The limiting dynamics as $\varepsilon \to 0$ consist of a family of two-dimensional, horizontally-incompressible, vertically-stratified Navier–Stokes equations parameterized by $x_3$. Specifically, for horizontal velocity $u_h(t, x_h, x_3)$: $$\begin{cases} \partial_t u_h + u_h \cdot \nabla_h u_h - \frac{1}{\mathrm{Re}} \Delta_h u_h = -\nabla_h p,\ \nabla_h \cdot u_h = 0,\ u_h|_{t=0} = u_{h,0}(x_h, x_3). \end{cases}$$ Here, $\nabla_h$ and $\Delta_h$ denote operators acting on the two horizontal coordinates $x_1, x_2$, while $x_3$ remains a parameter. Vertical dynamics are suppressed, corresponding physically to the averaging out of fast internal gravity waves at frequency $N$ and the emergence of columnar, stratified flow with full diffusive regularization in all directions. Global well-posedness for this regime has been rigorously established under general conditions (Scrobogna, 2017).

7. Physical and Mathematical Significance

The dimensionless Navier–Stokes equations, and especially their low-Froude-number limit, provide a foundation for the analysis of strongly stratified geophysical flows, such as atmospheric or oceanic dynamics, where the vertical scale is much smaller than the horizontal. The singular perturbation structure highlights the separation of scales in the dynamics, with direct implications for reduced modeling and the derivation of effective two-dimensional systems from fully three-dimensional Navier–Stokes–Boussinesq equations (Scrobogna, 2017). A plausible implication is that such scaling arguments remain central to further mathematical and computational studies of stratified turbulence and internal wave dynamics, especially in regimes where buoyancy forces dominate over inertial effects.