Ekman Spiral in Rotating Flows

• Ekman spiral is the depth-dependent turning of horizontal velocity in rotating viscous fluids, where wind stress and Coriolis force are balanced by vertical momentum transport.
• Mathematical models use primitive equations, piecewise-uniform eddy viscosity, and asymptotic expansions to derive explicit decay and rotation profiles under varying boundary conditions.
• The phenomenon influences transport processes such as Ekman pumping and cross-boundary mass flux, challenging the classical 45° deflection with varied observational outcomes.

Searching arXiv for recent and foundational papers on Ekman spiral, Ekman layers, and related rotating-flow formulations. The Ekman spiral is the depth-dependent turning of horizontal velocity that arises in a rotating viscous or turbulently diffusive boundary layer when Coriolis acceleration is balanced by vertical momentum transport. In the classical setting, it is the steady horizontal flow profile induced by wind stress and Coriolis rotation, with vertical structure determined by viscosity; near a solid boundary, this same Coriolis–viscous competition defines the Ekman layer and underlies Ekman transport and Ekman pumping (Binz, 26 Jul 2025). Across recent formulations, the spiral is no longer confined to the textbook constant-viscosity, flat-boundary case: it appears in primitive-equation dynamics, atmospheric boundary-layer closures, spherical-coordinate asymptotics, wave-driven surface flows, and parameterized reduced models, while several studies also clarify which rotating-flow phenomena persist when Ekman layers are removed altogether (Narasimhan et al., 2023).

1. Classical balance and canonical structure

In the finite-depth primitive-equation formulation, the Ekman spiral is a steady, horizontally homogeneous solution $v_E=v_E(z)$ of the balance

$\nu_z \partial_z^2 v_E = f\,v_E^\perp,$

subject to wind-stress forcing at the top boundary and prescribed geostrophic flow at the bottom boundary. The corresponding equilibrium is $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$, so the spiral is the vertical profile of horizontal velocity produced by the competition between vertical viscosity and Coriolis rotation (Binz, 26 Jul 2025).

An equivalent atmospheric formulation writes the steady, horizontally homogeneous Ekman momentum equations as

$$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$

Here the spiral emerges because the turbulent stress components vary with height, and the Coriolis terms rotate the velocity vector away from the near-surface direction toward the geostrophic direction aloft. In this representation, the outer Ekman layer and the inner surface layer are coupled through the vertical stress distribution rather than through a constant eddy viscosity alone (Narasimhan et al., 2023).

A compact mathematical statement of the classical boundary-value problem appears in the atmospheric $f$-plane model with piecewise-uniform eddy viscosity: $$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$ with $\psi=u+iv$. The complex formulation makes explicit that the spiral is simultaneously a decay and a rotation in height, with the real and imaginary parts representing the two horizontal velocity components (Stefanescu, 2024).

The same local mechanism survives beyond planar approximations. In a spherical asymptotic model for large-scale wind-driven ocean drift, the leading-order Ekman problem is

$\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$

with nonlinear wind-stress forcing at the surface and $W(z_0)=0$ at the lower boundary of the layer. Theorem 1 in that work shows that, for sufficiently regular positive eddy viscosity $m(z)$, the solution is unique, its magnitude decreases with depth, and its direction rotates with depth with opposite senses in the two hemispheres (Puntini et al., 6 Feb 2026).

2. Vertical closure, boundary conditions, and analytic variants

The modern literature treats the Ekman spiral less as a single closed-form profile than as a class of rotating boundary-layer solutions whose detailed structure depends on how vertical stress is modeled. The principal variants represented in recent work are summarized below.

Formulation	Vertical closure or forcing	Reported consequence
Primitive equations	Vertical viscosity $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$0, wind stress at top, geostrophic flow at bottom	Finite-depth Ekman spiral is the unique equilibrium under a smallness condition (Binz, 26 Jul 2025)
ABL analytical model	Self-similar total stress plus MOST-consistent surface layer	Spiral turning and low-level-jet structure in CNBL and SBL flows (Narasimhan et al., 2023)
Piecewise-uniform viscosity	Step-function $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$1 with jump matching	Explicit piecewise-exponential spiral solutions and variable surface angle (Stefanescu, 2024)
Damped Ekman theory	Linear damping term $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$2 added to Ekman balance	Flattened spiral with distinct amplitude and rotation depth scales (Wu et al., 4 May 2025)
Spherical asymptotics	Arbitrary depth-dependent eddy viscosity $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$3	Classical spiral behavior without an $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$4-plane approximation (Puntini et al., 6 Feb 2026)

In the atmospheric two-layer analytical model, the outer-layer stress magnitude follows the Nieuwstadt $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$5-power law,

$\nu_z \partial_z^2 v_E = f\,v_E^\perp,$6

while the inner-layer streamwise velocity follows a MOST-consistent form,

$\nu_z \partial_z^2 v_E = f\,v_E^\perp,$7

The cross-wind stress is then stitched to the outer-layer stress at a matching height $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$8 with $\nu_z \partial_z^2 v_E = f\,v_E^\perp,$9, yielding an analytic Ekman spiral that couples surface-layer similarity theory to outer-layer veering (Narasimhan et al., 2023).

In the piecewise-uniform eddy-viscosity problem, each layer admits a constant-coefficient solution, and matching requires continuity of velocity and continuity of flux across viscosity jumps. For one jump, the authors derive explicit coefficients for the piecewise-exponential solution; for finitely many jumps, they formulate a square linear system $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$0 and establish existence and uniqueness for the one-jump and two-jump cases, with an inductive extension for sufficiently small successive jumps (Stefanescu, 2024).

This suggests that the “Ekman spiral” is best treated as a structural consequence of rotating vertical momentum balance rather than as a single profile formula. Constant viscosity is one special closure; self-similar stress laws, stepwise viscosity, damping, and spherical geometry all preserve the turning-with-depth structure while modifying the quantitative spiral.

3. Surface angle, spiral flattening, and the non-universality of $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$1

A recurrent misconception is that the surface current or wind in an Ekman layer must be deflected by $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$2. The recent literature does not support that as a universal statement.

The damped theory begins from

$(u_E,\pi_E)=(v_E,0,-\rho_0gz)$3

and derives a surface-angle correction

$(u_E,\pi_E)=(v_E,0,-\rho_0gz)$4

For any nonzero damping, $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$5 and $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$6, while the spiral acquires two different depth scales, one for amplitude decay and one for directional rotation. The essential result is that $(u_E,\pi_E)=(v_E,0,-\rho_0gz)$7, so the speed decays more rapidly than the direction rotates; the observed spiral is therefore “flatter” than the classic constant-viscosity spiral (Wu et al., 4 May 2025).

The piecewise-uniform viscosity model reaches a different but complementary conclusion. In the one-jump case, the surface deflection angle depends on both the viscosity contrast and the jump height, and the authors show that while the constant-viscosity limit recovers

$(u_E,\pi_E)=(v_E,0,-\rho_0gz)$8

extreme viscosity contrasts can drive the angle to

$(u_E,\pi_E)=(v_E,0,-\rho_0gz)$9

The paper therefore demonstrates that the bottom-surface angle of the Ekman layer is not locked to $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$0 once the eddy viscosity varies with height (Stefanescu, 2024).

The spherical asymptotic model with explicit eddy-viscosity profiles sharpens the same point. Constant viscosity behaves most like the classical $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$1 case, linearly increasing viscosity yields surface deflection angles smaller than $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$2, while linearly decreasing and exponentially decaying viscosities often yield deflection angles larger than $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$3 (Puntini et al., 6 Feb 2026).

Taken together, these results indicate that $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$4 is a property of a very specific closure, not of the Ekman spiral as a general rotating-boundary-layer phenomenon. A plausible implication is that the most informative diagnostic of an Ekman spiral is not a fixed surface angle, but the combined vertical decay and turning implied by the chosen stress closure.

4. Transport, pumping, and cross-boundary mass flux

In the classical theory summarized in the damping study, the vertically integrated transport is perpendicular to wind stress and its curl produces Ekman pumping. The damped extension modifies both results. The transport becomes

$$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$5

so its magnitude is reduced by the factor $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$6 and it is no longer exactly perpendicular to wind stress. The corresponding pumping depends not only on wind-stress curl but also on wind-stress divergence; the paper emphasizes this as a major departure from the classic theory (Wu et al., 4 May 2025).

The same work gives explicit quantitative examples. Under a representative damping case with $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$7, the annual-mean zonally integrated Ekman transport along $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$8N is about $$0=f_c[V(z)-V_g]+\frac{\partial T_{xz}(z)}{\partial z},\qquad 0=-f_c[U(z)-U_g]+\frac{\partial T_{yz}(z)}{\partial z}.$$9 smaller, corresponding to about $f$0 below the classic prediction, with a maximum overestimation of about $f$1. For pumping, the correction can reach about $f$2 and exceed $f$3 in relative magnitude in some places near the equator (Wu et al., 4 May 2025).

On the tilted $f$4-plane, Ekman pumping is treated asymptotically by replacing the thin viscous layer with pumping boundary conditions. For no-slip boundaries, matching to the Ekman layer gives

$f$5

whereas for stress-free boundaries the pumping is weaker and depends on the vorticity gradient,

$f$6

The paper interprets the no-slip case as the tilted-$f$7-plane analogue of classical Ekman pumping and suction, with cyclonic vorticity pumping fluid away from the boundary and anticyclonic vorticity drawing fluid in (Tro et al., 2024).

In sloping, stratified boundary layers, the same transport logic becomes column-integrated. The transport-constrained slope model imposes

$f$8

typically with $f$9, so any cross-slope boundary-layer Ekman transport must be balanced by an interior return flow. Coriolis turning of that return flow then drives rapid along-slope adjustment. The paper frames this as a unified theory for Ekman arrest and spin down, with timescales

$$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$0

and controlling ratio $$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$1 (Peterson et al., 2022).

5. Spin-up, transient patterning, and instability of Ekman-type spirals

The Ekman spiral is not only a steady boundary-value problem; it also organizes transient spin-up and can itself become unstable.

In impulsively started rotating convection with a freely evaporating top surface, the flow evolves through a sequence

$$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$2

The relevant time scale is the Ekman spin-up time

$$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$3

and the ringed state appears at about $$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$4 and breaks down by $$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$5. Force extraction from PIV shows that during the ringed phase,

$$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$6

so Coriolis and viscous forces dominate inertia. The paper interprets the ringed state as an unsteady analogue of Ekman balance, and the subsequent breakdown as a loss of phase coherence between the Coriolis and viscous force distributions that triggers Kelvin–Helmholtz instability (Zhong et al., 2010).

A wave-driven analogue appears in the Ekman–Stokes spiral generated by surface gravity waves in a rotating frame. The Stokes drift profile $$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$7, together with rotation and a wave-induced surface stress, produces a horizontally invariant mean flow with Ekman-like turning in depth. This base flow is unstable above a critical Rossby number. In the large-$$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$8 limit the threshold tends to

$$(K\psi')'-2i(\psi-\psi_g)=0,\qquad \psi(0)=0,\qquad \lim_{z\to\infty}\psi(z)=\psi_g,$$9

recovering the standard Ekman-spiral instability driven by wave-induced surface stress rather than wind stress, whereas in the low-$\psi=u+iv$0 limit the threshold becomes

$\psi=u+iv$1

and the unstable mode depends crucially on the Stokes-drift profile (Seshasayanan et al., 2019).

Rotating shear flows exhibit a related instability landscape. In Ekman–Couette flow, the base state has two nonzero horizontal components,

$\psi=u+iv$2

with complex representation

$\psi=u+iv$3

External rotation can make the flow linearly unstable; at small rotation rates the onset scales as

$\psi=u+iv$4

and the optimal transient growth in the linearly stable region is slightly enhanced, approximately $\psi=u+iv$5. The optimal perturbations are inclined roll structures twisted by rotation, which the paper treats as a perturbation-level manifestation of Ekman-type turning in a bounded shear flow (Shi et al., 2013).

These results suggest that the Ekman spiral should be regarded not merely as a steady hodograph, but as a dynamically active shear configuration that can mediate spin-up, set transient pattern scales, and undergo linear or Hopf-type instability.

6. Geometry, reduced models, and domains of extension

Recent work shows that Ekman-spiral physics survives substantial changes in geometry, provided the local Coriolis–stress balance is preserved.

On the tilted $\psi=u+iv$6-plane, gravity and rotation are not aligned: $\psi=u+iv$7 The analysis therefore uses a non-orthogonal coordinate aligned with the rotation axis, not the geometric vertical. In this representation, the inner-layer equations reduce to the same fourth-order linear ODE structure as the classical upright Ekman layer, and the horizontal velocities in the inner layer have the familiar oscillatory-decaying form. Ekman layers scale like $\psi=u+iv$8, whereas anisotropic convective columns scale like $\psi=u+iv$9, so the tilted problem separates interior quasi-geostrophic dynamics from boundary-layer pumping while preserving the local spiral mechanism (Tro et al., 2024).

The large-scale spherical ocean model removes the tangent-plane approximation altogether. After a double asymptotic expansion in the thin-shell parameter and the Rossby-number parameter, the leading-order dynamics reduce to a vertical complex ODE whose solution behaves like a classical Ekman spiral for arbitrary eddy viscosity profiles. The current magnitude decreases with depth, the direction rotates with depth, and the turning sense is hemisphere-dependent; additionally, the surface deflection vanishes near the equator as $\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$0 (Puntini et al., 6 Feb 2026).

Wave-driven and superfluid extensions further broaden the scope of the concept. In the Ekman–Stokes problem, the forcing is not wind stress but a Stokes-drift-induced viscous surface stress, yet the large-$\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$1 limit collapses to the standard Ekman-spiral instability problem (Seshasayanan et al., 2019). In superfluid He II and neutron-star analogues, the papers emphasize a secondary circulation analogous to classical Ekman pumping, but the restoring mechanism is vortex-line tension rather than viscosity; the consequence is oscillatory rather than monotonic adjustment, with period

$\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$2

in the perfect-pinning limit (Eysden, 2015).

A plausible synthesis is that “Ekman spiral” names a family of rotationally constrained boundary-layer structures whose common invariant is turning-by-depth under Coriolis control, while the detailed profile, pumping law, and stability depend on geometry, closure, and forcing.

7. Absence of Ekman layers, reduced surrogates, and asymptotic fate

The significance of the Ekman spiral becomes especially clear in studies that remove or replace the Ekman layer.

In rotating convection with free-slip boundaries, the authors impose boundary conditions that eliminate the classical Ekman layers. Consequently there are no Ekman layers, no classical Ekman spiral boundary-layer structure, and no Ekman pumping from no-slip-wall boundary dynamics. Yet a three-regime organization still persists, classified by

$\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$3

with near-onset laminar scaling, an intermediate rotation-controlled regime, and a high-$\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$4 regime approaching non-rotating turbulent convection. The central interpretation is that a substantial part of the rotating-convection regime structure is due to bulk rotational constraint rather than Ekman-layer pumping (Schmitz et al., 2009).

At the opposite extreme, reduced quasi-geostrophic models often retain only the net dissipative effect of the boundary layer. In the two-layer QG model, the Ekman term enters asymmetrically as $\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$5 in the lower-layer dissipation. The paper is explicit that this term models friction with the surface boundary layer rather than the geometric spiral itself. Under the standard formulation it is always dissipative for energy, while under the extrapolated formulation there exist small negative regions in which it may inject energy or potential enstrophy. The mechanism is therefore Ekman-layer friction represented spectrally, not an explicit vertical turning profile (Gkioulekas, 2018).

For the full primitive equations with wind-driven boundary conditions, the most definitive asymptotic statement is the recent convergence theorem: under a quantitative smallness condition $\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$6, every strong solution converges exponentially fast to the Ekman spiral equilibrium,

$\bigl(m(z)W'(z)\bigr)'=2i\sin\theta\,W(z),$7

This implies that the Ekman spiral is the unique equilibrium of the system (Binz, 26 Jul 2025).

These results jointly delimit the role of the Ekman spiral. It is neither a universal explanation for all rotating-flow organization nor a dispensable textbook artifact. When the layer is removed, some rotational scalings survive; when it is reduced to a drag term, only its net frictional action remains; and when the full wind-driven primitive-equation dynamics are retained, the spiral emerges as the globally attracting equilibrium.