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 Incoming links: Buoyancy displacement variable

Use perturbations

v, b, p, ϑ

D_{0} = \partial_{t} - \frac{3}{2} Ω x \partial_{y} .

This is advection by the background Keplerian shearing sheet. Subtracting the equilibrium before retaining first-order terms gives the nine linearized equations:

D_{0} v_{x} - 2Ω v_{y} D_{0} v_{y} + \frac{1}{2} Ω v_{x} D_{0} v_{z} D_{0} b_{x} D_{0} b_{y} D_{0} b_{z} D_{0} ϑ \partial_{x} v_{x} + \partial_{y} v_{y} + \partial_{z} v_{z} \partial_{x} b_{x} + \partial_{y} b_{y} + \partial_{z} b_{z} = - ρ_{0}^{- 1} \partial_{x} p - N^{2} ϑ + \frac{B _{0}}{4 π ρ _{0}} \partial_{z} b_{x}, = - ρ_{0}^{- 1} \partial_{y} p + \frac{B _{0}}{4 π ρ _{0}} \partial_{z} b_{y}, = - ρ_{0}^{- 1} \partial_{z} p + \frac{B _{0}}{4 π ρ _{0}} \partial_{z} b_{z}, = B_{0} \partial_{z} v_{x}, = - \frac{3}{2} Ω b_{x} + B_{0} \partial_{z} v_{y}, = B_{0} \partial_{z} v_{z}, = v_{x}, = 0, = 0.

The azimuthal coefficient

Ω/2

combines the Coriolis acceleration with perturbation advection of the background shear. The

- 3Ω b_{x} /2

term is the winding of radial magnetic field by that shear. The tidal force cancels when the equations at a fixed position are subtracted; it does not add a separate Eulerian perturbation force. Magnetic pressure is already included in

p

, leaving only linear magnetic tension. In this normalization

ϑ

has dimensions of length because

D_{0} ϑ = v_{x}

Let

ζ = \nabla \times u

and

ω = ζ + 2Ω e_{z}

\partial_{t} u - u \times ω = - \nabla (\frac{P}{ρ _{0}} + \frac{1}{2} ∣ u ∣^{2} - \frac{3}{2} Ω^{2} x^{2}) - N^{2} θ e_{x} .

\partial_{t} ω - \nabla \times (u \times ω) = - N^{2} \nabla θ \times e_{x} .

Both

\nabla \cdot u

and

\nabla \cdot ω

vanish. The vorticity equation is consequently

\partial_{t} ω_{j} + u_{i} \partial_{i} ω_{j} - ω_{i} \partial_{i} u_{j} = - N^{2} (\nabla θ \times e_{x})_{j} .

T_{ij} = u_{i} ω_{j} - ω_{i} u_{j}

, use the flux convention

(\nabla \cdot T)_{j} = \partial_{i} T_{ij}

, with the first index denoting transport direction. This gives

\partial_{t} ω + \nabla \cdot T = - N^{2} \nabla θ \times e_{x} .

θ

is a buoyancy displacement variable, so dimensional consistency gives it units of length; the printed potential-temperature terminology presupposes this normalization. Its curl source directly forces only the

y, z

\partial_{t} ω_{x} + \partial_{i} (u_{i} ω_{x} - ω_{i} u_{x}) = 0.

Its volume integral is constant when the net boundary flux vanishes. However

D_{t} ω_{x} = ω \cdot \nabla u_{x}

still permits vortex stretching; the radial component is not generally a materially conserved scalar in three dimensions. Using divergence on the second index of the stated tensor would reverse the transport sign, so the tensor convention matters.