Boussinesq approximation

The Boussinesq approximation treats density as constant except where a small density variation multiplies gravity to produce buoyancy. The velocity remains incompressible.

In the constant-property convection convention, the strong Boussinesq approximation supplements small relative mass density changes with constant dynamic viscosity, thermal diffusivity, specific heat capacity and thermal expansion coefficient throughout the modeled temperature interval. A uniform reference density replaces density in inertia and mass conservation, while a linear density anomaly is retained in gravitational buoyancy. The temperature obeys an advection-diffusion equation with constant coefficients. The term “strong” is not uniform across all fluid-mechanics literature: some stratified-flow treatments instead emphasize the use of a globally constant reference density. The constant-property convection usage is explained in courses.physics.ucsd.edu/2021/Spring/physics218c/Spiegel%20Convection.pdf .

Uniformly stratified, nonrotating, two-dimensional ideal Boussinesq equations in coordinates $(x,z)$ are equivalent to unstratified rotating equations independent of $Y$, with $(X,Y,Z)=(z,y,-x)$, rotation $2\mathbf{\Omega }=N{\mathbf{e}}_Z$, velocity $(U,V,W)=(w,\sigma /N,-u)$, and the same reduced pressure. The transverse rotating velocity represents buoyancy perturbation, while the rotation-axis velocity represents the stratified horizontal velocity. Matching prescribed forces may have components in the invariant plane, but no transverse component. Physical free surface conditions are not preserved: the physical vertical direction and its normal velocity are different under the mapping.

A weak horizontal force $f^{\prime }(x)e^{imz}$, with $m>0$ and $\int f^{\prime }(x)dx=\epsilon$, admits many steady pressures differing by an arbitrary function of height. Switching the force on in initially motionless, unbounded, uniformly stratified fluid selects $p_{\infty }=[f(x)-\epsilon /2]e^{imz}$, up to a spatially constant pressure gauge. For horizontal wavenumber $q$, the transient frequency is $\Omega (q)=Nq/\sqrt{q^2+m^2}$. Its long-wave group velocity is $N/m$, and oppositely propagating outgoing fronts leave pressure limits $-\epsilon /2$ and $+\epsilon /2$. The limit is at fixed $x$ after the fronts have passed; dispersive tails need not vanish at any finite time.

For a weak horizontal forcing proportional to $e^{imz}$ in an ideal uniformly stratified nonrotating fluid, the low-frequency hydrostatic approximation gives the displayed one-dimensional wave equation for the streamfunction amplitude. Two oppositely propagating characteristic integrals determine the response. A source confined to $|x|\le a$ and switched on at time zero produces no response beyond $|x|\le ct+a$ in this reduced equation. The bound on mixed forcing derivative $|f_{xt}|\ll |mN{f}_{\max }|$ controls the neglected vertical acceleration in characteristic magnitude.

With mechanically and thermally equivalent upper and lower boundaries, the Boussinesq approximation permits reflection through the layer midplane together with reversal of the temperature perturbation and vertical velocity. In a reduced convection temperature equation this induces temperature inversion. A quadratic term which changes differently from the linear terms breaks this symmetry; a cubic term can preserve it. The Boussinesq approximation alone does not impose symmetry on unequal boundary conditions.

The inviscid Boussinesq equations for velocity $\mathbf{u}$, buoyancy $b$, and reference density ${\rho }_0$ aretogether with an advection or advection–diffusion equation for $b$.

The Linearized Boussinesq equations describe small velocity, pressure, and buoyancy perturbations about a motionless, stably stratified reference state. With kinematic viscosity $\nu$, mass diffusivity $\kappa$, and buoyancy frequency $N$, they are

An inertial wave is a rotating-fluid wave restored by the Coriolis force. Its frequency depends on the angle between its wavevector and the rotation axis.

For fixed vertical wavenumber $k_z$, the inertial-wave dispersion relation is ${\omega }^2=4{\Omega }^2{k}_z^2/(k_x^2+k_z^2)$. Its horizontal group velocity is the derivative displayed above, opposite in sign to $\omega k_x$. In a half-channel $x>0$, an outgoing propagating mode therefore requires $\omega k_x<0$. With time factor $e^{-i\omega t}$ and $q=|k_z|\sqrt{4{\Omega }^2/{\omega }^2-1}$, choose $e^{-i\mathrm{sgn}(\omega )qx}$. Above the inertial-frequency cutoff, the horizontal wavenumber is imaginary and the appropriate radiation condition instead selects the exponentially decaying solution. Energy transport, rather than the direction of phase travel alone, determines the outgoing sign.

Linear incompressible motion about rest in a frame rotating with $\mathbf{\Omega }=\Omega \mathbf{k}$ obeys ${\mathbf{u}}_t+2\mathbf{\Omega }\times \mathbf{u}=-\nabla P/\rho$. If $\zeta =v_x-u_y$ is its vertical vorticity, taking curl gives ${\zeta }_t=2\Omega w_z$, while eliminating pressure using continuity gives ${\partial }_t{\nabla }^{2}w+2\Omega {\zeta }_z=0$. Differentiating in time yields the displayed scalar equation. All three velocity components must be retained, even when the fields depend on only two spatial coordinates. The Coriolis acceleration generally generates a transverse velocity that is absent in a strictly two-component description.

For $\omega =\sigma 2\Omega k_{\parallel }/k$, differentiating the dispersion relation gives the displayed group velocity. It is perpendicular to the wavevector. If $|k_{\parallel }|\ll k_{\perp }$, energy propagates almost along the rotation axis while the phase velocity is almost horizontal. In an orthonormal basis ${\mathbf{e}}_1,{\mathbf{e}}_2$ normal to $\mathbf{k}$, with ${\mathbf{e}}_2=\hat{\mathbf{k}}\times {\mathbf{e}}_1$, the complex velocity is $A({\mathbf{e}}_1-\sigma i{\mathbf{e}}_2)$ and its real oscillation is circular. In the anisotropic limit the vertical and transverse-horizontal velocity amplitudes are comparable, while the horizontal component parallel to ${\mathbf{k}}_{\perp }$ is smaller by $|k_{\parallel }|/k_{\perp }$.

Insert a transverse plane wave $\mathbf{u}={\mathbf{u}}_0{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$ into the inertial-wave vorticity equation, with $\mathbf{k}\cdot {\mathbf{u}}_0=0$. It gives $\omega \mathbf{k}\times {\mathbf{u}}_0=2i(\mathbf{\Omega }\cdot \mathbf{k}){\mathbf{u}}_0$. Applying $\mathbf{k}\times$ again and using transversality gives the title dispersion relation. If $\theta$ is the angle to the rotation axis, $|\omega |=2|\Omega \cos \theta |$. At fixed direction the angular frequency does not depend on wavelength. Parallel wavevectors give the maximum $2|\Omega |$; perpendicular ones have zero frequency and belong to the nonoscillating balanced limit.

Linearize the Euler equation in a uniformly rotating frame about rest and impose incompressible flow, $\nabla \cdot \mathbf{u}=0$. Taking the curl eliminates the reduced pressure. For constant $\mathbf{\Omega }$, $\nabla \times (\mathbf{\Omega }\times \mathbf{u})=-(\mathbf{\Omega }\cdot \nabla )\mathbf{u}$, giving the displayed vorticity equation. Taking another curl and time derivative yields ${\partial }_t^2{\nabla }^2\mathbf{u}+4(\mathbf{\Omega }\cdot \nabla {)}^2\mathbf{u}=0$. These equations describe the restoring influence of background rotation on small disturbances without buoyancy or viscosity.

In a uniform-density incompressible flow inside a smooth rigid sphere rotating at angular speed $\Omega$, a normal mode with phase $e^{i\sigma t+im\varphi }$ has intrinsic frequency $s=\sigma +m\Omega$. For $s\ne 0,\pm 2\Omega$, eliminating velocity from the linearized Euler equations gives the displayed pressure equation, with $W=p^{\prime }/\rho$. Impermeability gives $R{W}_R+(2m\Omega /s)W+(1-4{\Omega }^2/s^2)z{W}_z=0$ on the sphere. There is no condition $W=0$: the rigid wall supplies variable normal pressure. These inertial waves are restored by Coriolis acceleration, and their pressure equation is hyperbolic for $|s|<2|\Omega |$.

For an axisymmetric normal mode in a rigid rotating sphere, substitute $W=z(A{R}^2+B{z}^2+C)$ into the pressure equation and its impermeability condition. With $\beta =1-4{\Omega }^2/{\sigma }^2$, the interior equation gives $2A+3\beta B=0$. The boundary polynomial gives $3\beta B=(2+\beta )A$ and $\beta C=-(2+\beta )A{r}_0^2$. A nontrivial oscillatory solution has $\beta =-4$, hence the displayed frequency and $B=A/6$, $C=-A{r}_0^2/2$. It satisfies both equations identically.