Four real fields in a rotating patch of a disk

Prerequisites are circular motion, force, derivatives as rates of change, and sine and cosine. The preceding angular-momentum explanation supplies the physical question: when does a magnetic connection amplify a displacement? By the end, you should be able to reconstruct the four equations, check a proposed initial state and explain why a pattern varying vertically need not have vertical fluid velocity.

Define the patch before removing the orbit

Choose a reference circular orbit with radius $R_0$ and positive angular speed $\Omega$, in radians per second. The local frame follows that orbit. Its $x$ direction points radially outward, $y$ follows the orbital motion, and $z$ points along the uniform background magnetic field $B_0>0$. Distances are small compared with $R_0$, so we keep the first radial variation of angular velocity. Define

The parameter $q$ is dimensionless. The background velocity measured in this rotating frame is

To obtain this expression, expand the neighbouring angular speed to first order, then multiply its difference from the frame speed by $R_0$. For positive $q$, outer gas falls behind the frame and inner gas moves ahead. For a Keplerian rotation law, $q=3/2$. Stone and Gardiner's pinned Section II, equations 7–8, gives this convention.

The fluid has uniform mass density $\rho$, measured in kilograms per cubic metre. It is an ideal conductor, is incompressible and has no vertical stratification. The perturbation is independent of $x$ and $y$, and repeats after a vertical length $L$. Buoyancy, resistivity and nonlinear products of perturbations are omitted. This model follows a local linear channel; a turbulent disk requires the nonlinear terms.

Give the magnetic disturbance velocity units

Write a magnetic disturbance in Alfvén velocity units,

Here $\mu_0$ is the SI magnetic permeability. Both $\delta\mathbf b$ and $v_A$ have units of velocity. This absorbs a magnetic normalization into the variables; it does not identify magnetic field with fluid velocity. Balbus and Hawley use Gaussian magnetic units in their original component equations. The normalization above lets us use the same physical coupling in SI.

Choose a positive integer $n$ and vertical wavenumber $k=2\pi n/L$. Let velocity disturbances multiply sine and magnetic disturbances multiply cosine:

The four symbols $u_x,u_y,b_x,b_y$ now denote amplitudes that depend on time. Their units are metres per second. The background field remains vertical and constant. The disturbances have no vertical component. Their divergences vanish because horizontal components have no horizontal variation; a derivative of a horizontal component with respect to height does not enter the divergence.

Follow one spatial derivative to fix the signs

Magnetic tension in the radial momentum equation contributes $v_A\partial_z\delta b_x$. Since

its amplitude is negative, $-kv_A b_x$. In the induction equation, stretching the vertical field contributes $v_A\partial_z\delta v_x$, with positive amplitude $kv_Au_x$. The apparent sign difference comes from the quarter-wavelength phase offset.

The rotating frame adds Coriolis terms. Radial motion also carries a parcel across the background shear. Together they give $2\Omega u_y$ in the radial equation and $-(2-q)\Omega u_x$ in the azimuthal equation. Finally, the background shear turns a radial magnetic disturbance into an azimuthal disturbance at rate $-q\Omega b_x$. This is the magnetic counterpart of drawing a slanted line across two streams with different speeds.

Define the tension frequency $K=kv_A$, in inverse seconds. The four amplitude equations are

A dot means a derivative with respect to time. Every right-hand term has velocity divided by time. These equations are the unstratified, vertical-wave specialization of the original linear component equations 2.2; no complex-valued amplitude is needed for this phase convention.

Check one actual seed by substitution

The saved fastest-mode case uses $q=3/2$, $\Omega=1/T_0$, $K=\sqrt{15}/(4T_0)$, and

Its box length is $L_0$, density is one in the declared units, and the fundamental pattern has $k=2\pi/L_0$. Thus its background Alfvén speed is $v_A=KL_0/(2\pi)$. The field perturbation is initially small compared with that background.

For the radial velocity, the two terms are $2A/T_0$ and $-5A/(4T_0)$. Their sum is $3A/(4T_0)$. The azimuthal equation gives the same derivative. The magnetic equations give $3/(4T_0)$ times their respective amplitudes. All four components therefore initially grow in their existing proportions. The next chapter derives why this is the fastest Keplerian mode.

Compare this algebra with the actual frame. At the saved time $t=3T_0$, the fundamental radial velocity amplitude is approximately $9.48817\times10^{-4}L_0/T_0$. The continuum exponential reference gives $9.48774\times10^{-4}L_0/T_0$. The numerical field uses a spatial mesh; the reference is a separate analytic solution. The plot keeps the saved field, including its finite-resolution differences.

Remove one coupling and predict the result

If the field coupling vanishes, $K=0$, the magnetic variables no longer push the velocity. Differentiate the radial equation and use the azimuthal one:

The frequency $\kappa$ describes a small epicyclic oscillation around the reference orbit. For Keplerian shear and an initial radial kick, the solution is $u_x=A\cos(\Omega t)$, $u_y=-A\sin(\Omega t)/2$. This independent limit checks the signs of the rotation terms. The saved zero-field case was compared with this explicit oscillating solution.

Exercise1. A graph shows a sinusoidal horizontal velocity that varies with height. Must it have vertical fluid motion?

Answer. No. Height labels where the horizontal motion is measured. In this channel, the vertical velocity is zero everywhere. Check incompressibility using the derivative of each velocity component along its own coordinate direction.

Exercise2. Someone uses cosine for both velocity and magnetic disturbances but keeps exactly these four real equations. What is missing?

Answer. A vertical derivative changes cosine into sine. The proposed four-variable set no longer contains the spatial pattern produced by the derivative. One must add the missing quadratures or use a consistent complex representation. Changing phase notation cannot justify keeping inconsistent signs.

Exercise3. Set $q=0$ without removing rotation. Is this the same as setting $\Omega=0$?

Answer. Rigid rotation still has Coriolis coupling and epicyclic motion. A nonrotating magnetic fluid has only the Alfvén coupling in these equations. These are different useful controls, and the saved spatial calculation includes both.

Sources

Balbus and Hawley (1991), printed page 216, equations 2.2a–g, supplies the linear component system; pages 216–218, equations 2.5–2.9 and2.13–2.14b, state its dispersion and stability context. Primary scan. Stone and Gardiner, arXiv:1006.0139v1, Section II equations 1–8, fixes the rotating-frame conventions and magnetic normalization; equations 11–13 provide an epicyclic check. Pinned version. The displayed spatial frame is an existing linear periodic-field record. No new nonlinear evolution is implied.