Anelastic Vorticity Equation

Recall the anelastic momentum equation is

\[\begin{equation} \frac{D \mathbf{v}}{Dt} + f\mathbf{k}\times \mathbf{v} = -\nabla \left(\frac{p}{\overline{\rho}}\right) + g\delta \phi \mathbf{k}, \label{eq:anmom} \end{equation}\]

where \(\mathbf{v}=(u, v, w)\).

Let \(\nabla \mathbf{v}\) denote the matrix (i.e. tensor) with columns one, two, and three containing the gradient of \(u, v, w\), respectively. Note

\[\begin{split}\begin{align*} \mathbf{v} \cdot \nabla\mathbf{v} &= \left(u,v,w\right) \begin{pmatrix} u_x & v_x & w_x \\ u_y & v_y & w_y \\ u_z & v_z & w_z \\ \end{pmatrix}\\ &= \left(uu_x+vu_y+wu_z, uv_x+vv_y+wv_z, uw_x+vw_y+ww_z\right) \\ &= \nabla\frac{|\mathbf{v}|^2}{2}+ \begin{pmatrix} vu_y+wu_z-vv_x-ww_x, \\ uv_x+wv_z-uu_y-ww_y, \\ uw_x+vw_y-uu_z-vv_z \\ \end{pmatrix}^T \\ &=\nabla\frac{|\mathbf{v}|^2}{2}+ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ w_y-v_z & u_z-w_x & v_x - u_y \\ u & v & w \end{vmatrix} \\ &=\nabla\frac{|\mathbf{v}|^2}{2}+ \boldsymbol{\omega}\times\mathbf{v} \end{align*}\end{split}\]

Following Markowski (2010, p. 21), to derive the vorticity equation, take the curl of both sides of equation \(\eqref{eq:anmom}\) and apply vector identities to obtain

\[\begin{split}\begin{align} \nabla\times\left(\frac{D \mathbf{v}}{Dt} + f\mathbf{k}\times \mathbf{v}\right) &= -\nabla \left(\frac{p}{\overline{\rho}}\right) + g\delta \phi \mathbf{k} \\ \frac{D(\boldsymbol{\omega} + f \mathbf{k})}{Dt} + (\boldsymbol{\omega} + f \mathbf{k})(\nabla\cdot \mathbf{v}) &= \left[\left(\boldsymbol{\omega} + f \mathbf{k}\right)\cdot \nabla\right]\mathbf{v} + \nabla \times \left(g\delta \phi \mathbf{k}\right), \end{align}\end{split}\]

where \(\boldsymbol{\omega} = (\xi, \eta, \zeta)\) is the vorticity vector, and \(\nabla \mathbf{v}\) is a tensor. Considering just the \(y\) component,

\[\frac{D\eta}{Dt} + \eta\left[u_x + v_y + w_z\right] = \left(\boldsymbol{\omega}+f\mathbf{k}\right)\cdot\nabla v + g\phi_x,\]

where \(x, y, z\) subscripts denote partial derivatives. The continuity equation implies

\[-w \eta \frac{1}{\overline{\rho}}\frac{\partial \overline{\rho}}{\partial z} = \eta\left[u_x + v_y + w_z\right].\]

Substituting into the previous equation and multiplying through by \(\frac{1}{\overline{\rho}}\) gives

\[\begin{equation} \frac{D}{Dt}\left(\frac{\eta}{\overline{\rho}}\right) = \underbrace{\frac{1}{\overline{\rho}}\left(\boldsymbol{\omega}+f\mathbf{k}\right)\cdot\nabla v}_{\text{deformation}} + \underbrace{\frac{g}{\overline{\rho}}\frac{\partial \phi}{\partial x}}_{\text{generation}}. \label{eq:vort} \end{equation}\]

The quantity \(\frac{\boldsymbol{\omega}}{\overline{\rho}}\) is sometimes called “specific-vorticity” (Petropoulos et al., 2017). Equation \(\eqref{eq:vort}\) says that the horizontal component of a parcel’s (anelastic) specific vorticity can change only through vorticity deformation or baroclinic vorticity generation. In the absence of deformation and generation, if a parcel’s specific volume increases, its horizontal vorticity \(\eta\) must decrease, behavior analogous to that of angular momentum.