Parker spiral

Builds on Alfvén's frozen-in flux theorem Parker solar wind model

Definition

Let a star rotate with constant angular velocity $\Omega > 0$ about the $z$ -axis, and use spherical coordinates $(r, \theta, \phi)$ in an inertial frame centred on the star. Assume that beyond a sphere of radius $r_0$ the plasma moves radially with a constant speed $v > 0$ , as in the Parker solar wind model. Assume that the magnetic field is frozen into the plasma. Let a field line be anchored at the point of the sphere $r = r_0$ with colatitude $\theta_0$ that has longitude $\phi_0 + \Omega t$ at time $t$ . The Parker spiral through this footpoint at time $t$ is the curve $\theta = \theta_0, \qquad \phi = \phi_0 + \Omega t - \frac{\Omega}{v} (r - r_0), \qquad r \ge r_0.$

Remarks

The plasma that sits at radius $r$ at time $t$ left the footpoint at time $t' = t - (r - r_0)/v$ , when the footpoint had longitude $\phi_0 + \Omega t'$ ; this gives the formula. At fixed $t$ , $r$ is an affine function of $\phi$ , so the curve is an Archimedean spiral, and its turns are spaced by $2\pi v/\Omega$ in $r$ . The field components are $B_r = B_0 (r_0/r)^2$ and $B_\phi = -B_r \, \Omega (r - r_0) \sin\theta / v$ , so the angle $\psi$ between the field and the radial direction satisfies $\tan\psi = \Omega (r - r_0) \sin\theta / v$ .

Example: for the Sun, $\Omega \approx 2.7 \times 10^{-6}$ per second, and at the Earth $r \approx 1.5 \times 10^{8}$ kilometres with $v \approx 400$ kilometres per second. Then $\Omega r / v \approx 1$ , so $\psi \approx 45$ degrees in the equatorial plane.

Non-example: a logarithmic spiral $r = r_0 e^{b\phi}$ is not a Parker spiral, because the distance between its successive turns grows with $r$ instead of staying equal to $2\pi v/\Omega$ .

Reading path

To understand this page from the ground up, read these 96 pages in order. The foundations come first.

Critical point concept
Derivative concept
Differential equation concept
Euclidean space object
Euclidean vector concept
Euler equations concept
Function concept
Function composition concept
Ideal MHD equations concept
Infimum concept
Interval concept
Magnetohydrodynamics concept
Ordinary differential equation concept
Speed of sound concept
Supremum concept
Velocity concept
Fréchet derivative definition
Total derivative concept
Cross product definition
Ordinary differential equation definition
Function of several variables concept
Vector-valued function concept
Lipschitz condition definition
Classical mechanics concept
Jordan measure definition
Differentiable function of several variables concept
Limit of a function of several variables concept
Momentum concept
Parametrised curve concept
Vector field concept
Iterated integral definition
Partial derivative concept
Solution of an ordinary differential equation definition
Inertial frame of reference concept
Continuity in R^n definition
Differentiable function of several variables definition
Gradient concept
Partial derivative definition
Curl concept
Divergence concept
Initial value problem definition
Newton's laws of motion statement
Differentiable on an open set definition
Double integral definition
Chain rule in several variables statement
Higher partial derivatives definition
Gradient definition
Jacobian matrix definition
Picard–Lindelöf hypotheses definition
Iterated derivative definition
Fubini's theorem statement
Chain rule for a composition with a curve statement
Triple integral definition
Picard operator definition
C^k function characterised by continuous derivatives statement
An iterate of the Picard operator is a contraction statement
Integral equation form of an initial value problem statement
Differential form definition
Parametrised curve definition
Vector field definition
Picard–Lindelöf theorem statement
Wedge product of forms definition
Closed curve definition
Rectifiable curve definition
Parametrised surface definition
Conservative vector field definition
Curl definition
Divergence definition
Tangent vector of a curve definition
Classical mechanics definition
Line integral of a scalar field definition
Line integral of a vector field definition
Tangent plane and normal vector of a surface definition
Momentum definition
Potential energy definition
Regular curve definition
Green's theorem statement
One-form definition
Angular momentum definition
Surface area definition
Arc length parametrisation definition
Divergence theorem on a rectangle in the plane statement
Surface integral definition
Exterior derivative definition
Line integral as an integral over the parameter interval statement
Central force definition
Orientable surface definition
Gradient theorem statement
Newton's law of universal gravitation statement
Flux through a surface definition
Parker solar wind model statement
Divergence theorem statement
Generalised Stokes theorem statement
Stokes' theorem statement
Alfvén's frozen-in flux theorem statement
Parker spiral definition

Tag	`WQRDJYHEW5`
URI	`m2://WQRDJYHEW5`
Kind	definition
Writing	W0 stub
MSC	85A30, 76W05
Defines	Parker spiral
Uses	Alfvén's frozen-in flux theorem, Parker solar wind model
Changed	3 November 2026

Parker spiral

Definition

Remarks

Reading path

Further reading

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