The Boussinesq approximation provides a fundamental framework for analyzing atmospheric and oceanic flows where density variations are small but buoyancy effects remain significant. Under this approximation, density variations are neglected everywhere except in the gravitational term of the vertical momentum equation (Vallis, 2017). The governing equations in Cartesian coordinates $(x, y, z)$ with velocity components $(u, v, w)$ are:
Here, $\mathbf{V} = (u, v, w)$ is the velocity vector, $p$ is the pressure perturbation, $\rho_0$ is the reference density, $b = -g(\rho - \rho_0)/\rho_0$ is the buoyancy with gravitational acceleration $g$, $\mathbf{F} = (F_u, F_v, F_w)$ represents frictional forces (including viscous and subgrid-scale effects), and $S_b$ accounts for buoyancy sources and sinks such as diabatic heating.
The kinetic energy density is defined as $KE = \frac{1}{2}\rho_0|\mathbf{V}|^2 = \frac{1}{2}\rho_0(u^2 + v^2 + w^2)$. To derive its evolution equation, we multiply equations (1a-c) by $\rho_0 u$, $\rho_0 v$, and $\rho_0 w$ respectively, then sum them.
The left-hand side of equation (1a) multiplied by $\rho_0 u$ gives:
Using the identity $u\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial u^2}{\partial t}$ and similar manipulations for the advective terms:
Similar expressions arise for the $v$ and $w$ components. After considerable algebra and using the continuity equation (1d), the complete kinetic energy equation becomes:
This is the kinetic energy equation in flux form, where the terms on the left represent the local time rate of change and the divergence of kinetic energy flux (including pressure work), while the right-hand side contains the buoyancy production and frictional dissipation terms.
Integrating equation (2) over the entire fluid domain $\mathcal{V}$ and applying the divergence theorem, the flux terms vanish at appropriate boundaries (assuming no-flux or periodic boundary conditions). The volume-integrated kinetic energy equation becomes:
where $[Q] \equiv \int_{\mathcal{V}} \rho_0 Q \, dV$ denotes the mass-weighted volume integral. This fundamental result shows that:
Consider an orthonormal decomposition of the velocity and buoyancy fields. For a domain with appropriate boundary conditions, we can expand in Fourier modes:
where $\mathbf{k} = (k_x, k_y, k_z)$ is the wavenumber vector. For planetary atmospheres, spherical harmonic expansions are more appropriate (Holton & Hakim, 2013).
The kinetic energy associated with each mode is $KE_{\mathbf{k}} = \frac{1}{2}\rho_0|\hat{\mathbf{V}}_{\mathbf{k}}|^2$. Setting aside transport terms temporarily, the evolution equation for each mode follows the same structure as equation (3):
where $*$ denotes complex conjugation and Re[$\cdot$] takes the real part. This demonstrates that the fundamental energy balance—buoyancy production and frictional dissipation—applies at each scale independently when transport effects are neglected.
The nonlinear advection terms, when written in advective form, are:
These nonlinear terms are responsible for energy transfer across different scales and wavenumbers, fundamentally altering the simple modal picture described above.
It is crucial to distinguish between wavenumber and scale as conceptual frameworks:
This distinction becomes important when discussing energy cascades: energy typically flows from large scales (low $k$) to small scales (high $k$) in three-dimensional turbulence, following Kolmogorov's theory (Frisch, 1995).
Frontogenesis provides a clear example of how large-scale flows generate small-scale features. The frontogenesis function $F$ measures the rate of strengthening of temperature or buoyancy gradients:
When $F > 0$, the horizontal strain field acts to intensify buoyancy gradients, creating increasingly sharp frontal zones. This process transfers energy from large scales (the synoptic-scale strain field) to progressively smaller scales (the sharpening front), ultimately requiring viscous dissipation or numerical diffusion to prevent infinite gradients (Hoskins, 1982).
In the spectral representation, the nonlinear terms lead to interactions between different Fourier modes. A fundamental result from weakly nonlinear theory is that energy exchange occurs primarily through resonant triads—sets of three wavenumbers $\mathbf{k}_1$, $\mathbf{k}_2$, and $\mathbf{k}_3$ satisfying the resonance condition:
where $\omega_i$ are the corresponding frequencies (Nazarenko, 2011). These interactions can either:
The resonant triad mechanism provides the theoretical foundation for understanding how energy initially input at large scales (e.g., by baroclinic instability in the atmosphere) cascades through the spectrum to small scales where it is ultimately dissipated.
The kinetic energy framework developed here has profound implications for understanding atmospheric circulation. The balance between buoyancy production $[bw]$ and frictional dissipation $[\mathbf{F}\cdot\mathbf{V}]$ governs the overall energy budget of atmospheric motions. Meanwhile, the nonlinear transport terms redistribute this energy across scales, from the energy-containing range (synoptic scales, ~1000 km) through the inertial subrange down to the viscous dissipation scales (~mm).
Understanding these processes is crucial for:
Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.
Holton, J. R., & Hakim, G. J. (2013). An Introduction to Dynamic Meteorology (5th ed.). Academic Press.
Hoskins, B. J. (1982). The mathematical theory of frontogenesis. Annual Review of Fluid Mechanics, 14(1), 131-151.
Nazarenko, S. (2011). Wave Turbulence. Springer-Verlag.
Vallis, G. K. (2017). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation (2nd ed.). Cambridge University Press.