1. Linear theory – General form

It is assumed that the velocity \((u_0,v_0,0)\) and the Brunt–Väisälä or buoyancy frequency \(N\) of the background state are independent of height, and that the Boussinesq approximation can be made. In the absence of rotation and friction, the governing equations for small perturbations \((u_1,v_1,w_1)\) in velocity, \(p_1\) in pressure and \(\theta_1\) in potential temperature are

(1.1)\[ \frac{\mathrm{D}u_1}{\mathrm{D}t} =-\frac{1}{\rho_0}\frac{\partial p_1}{\partial x}, \]

(1.2)\[ \frac{\mathrm{D}v_1}{\mathrm{D}t} =-\frac{1}{\rho_0}\frac{\partial p_1}{\partial y}, \]

(1.3)\[ \frac{\mathrm{D}w_1}{\mathrm{D}t} =-\frac{1}{\rho_0}\frac{\partial p_1}{\partial z} + \frac{\theta_1}{\theta_0}g, \]

(1.4)\[ \frac{\mathrm{D}\theta_1}{\mathrm{D}t} + w_1\frac{\mathrm{d}\theta_0}{\mathrm{d}z} =0, \]

(1.5)\[ \frac{\partial u_1}{\partial x}+\frac{\partial v_1}{\partial y}+\frac{\partial w_1}{\partial z}=0 \]

with the material derivative \(\frac{\mathrm{D}}{\mathrm{D}t} = \left(\frac{\partial}{\partial t}+\boldsymbol{u}_0\cdot\boldsymbol{\nabla}\right)\). Equations (1.1)-(1.5) can be reduced to a single equation in \(w_1\) as follows. Taking the material derivative of the vertical momentum equation (1.3) allows the substitution of the potential temperature equation (1.4). The pressure is found by taking the divergence of the momentum equations (1.1)-(1.3) and applying the continuity equation (1.5). This yields

(1.6)\[ \left(\frac{\mathrm{D}}{\mathrm{D}t}\right)^2\nabla^2 w_1+N^2\nabla^2_H w_1=0. \]

with the horizontal Laplacian operator and Brunt–Väisälä frequency defined as \(\nabla^2_H=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\) and \(N^2=\frac{g}{\theta_0}\frac{\mathrm{d}\theta_0}{\mathrm{d}z}\), respectively. Equation~(1.6) is a simplified form of the more general Taylor-Goldstein equation for wave motions in a stably stratified shear flow.

Equation (1.6) can also be expressed in terms of the vertical displacement \(\eta_1\) of a fluid parcel above its undisturbed level, which is related to the vertical wind speed perturbation via the kinematic condition [Smi80]

(1.7)\[ w_1=\frac{\mathrm{D}\eta_1}{\mathrm{D}t}. \]

Substituting equation (1.7) into equation (1.6) gives

(1.8)\[ \left(\frac{\mathrm{D}}{\mathrm{D}t}\right)^2\nabla^2 \eta_1+N^2\nabla^2_H \eta_1=0. \]

We can find a solution for equation (1.8) by assuming a plane wave solution of the form

\[ \eta_1(x,y,z,t)=\hat{\eta}_1(z)\exp{[j(kx+ly-\omega t)]}. \]

Substituting the planar wave form into (1.8) results in

(1.9)\[ \frac{\partial^2\hat{\eta}_1}{\partial z^2}+m^2\hat{\eta}_1=0 \]

with the vertical wave number \(m\) given by

(1.10)\[ m^2=(k^2+l^2)\left(\frac{N^2}{\Omega^2}-1\right). \]

The intrinsic frequency is hereby defined as \(\Omega=\omega-\mathbf{u_0}\cdot\mathbf{k}\), with \(\mathbf{k}=(k,l,m)\) the wave vector and \(\mathbf{u_0}=(u_0,v_0,0)\) the background wind speed vector.

Equation (1.9) is a linear, homogeneous, ordinary differential equation of second order, for which the general solution can be written as

(1.11)\[ \hat{\eta}_1(k,l,z,\omega) = A(k,l,\omega)e^{jm_1(k,l,\omega)z}+B(k,l,\omega)e^{jm_2(k,l,\omega)z}, \]

with \(m_1\) and \(m_2\) the roots of equation (1.10). The coefficients \(A\) and \(B\) are determined by the boundary conditions.