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Ice-Ocean Stress

I. Quadratic drag laws: background, physicsa. 2-d vectors as complex numbersb. Free-drift force balancec. Drag coefficients

II. IOBL similaritya. Dimensionless variablesb. Ekman stress equationc. Ekman “surface” velocityd. Combine Ekman and surface layerse. Rossby similarity-- why?

III. Impact of stratificationa. Rapid melting-- generalized Rossby similarityb. Ice edge bandsc. Shallow mixed layers

IV. Recommendationsa. Model considerationsb. Undersurface hydraulic roughness

McPhee, M. G., 2011: Advances in understanding ice-ocean stress during and since AIDJEX, ColdReg. Sci. Technol., doi:10.1016/j.coldregions.2011.05.001.

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2-D vectors as complex numbers

x (u)

y (v)

θ

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Free drift force balance: −ifmiceVice =ρaτa −ρwτw

(τa is kinematic air stress; τw is kinematic stress exerted by the

ice on the water column)

Force Balance 1:typical of AIDJEXz0 = 0.07 mIce Thickness: 3 mc10 = 0.0018|Vice|/Uwind=1.9%cw = 0.0055ei22°

Force Balance 2:typical of ANZFLUXz0 = 0.002 mIce Thickness: 0.5 mc10 = 0.0018|Vice|/Uwind=3.3%cw = 0.0020ei13°

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AIDJEX Pilot, 19725 h average on 12 Apr(McPhee and Smith, JPO, 1976

8 h average on 1 Apr

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Steady, horizontally homogeneous boundary layer equation:

ifv =∂τ∂z

=τ zzτ =Kvz

Constitutive law relating stress in the fluid to shear (Ekman postulated that eddy viscosity was independent of depth):

Boundary conditions: stress at surface specified, velocity vanishes at depth:

2nd order ordinary differential equation:

vzz −ifK

v=0

The Ekman Layer

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Ekman’s solution for the steady, unstratified boundary layer forced by stress at the surface of a deep ocean. The net volume transport is perpendicular to the surface stress

Solution for τ 0 =iτ 0

v(z)=τ 0

(1+ i)2 fK

exp[ f / (2K )(1+ i)z]

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−2 m < zsl < 0

Solution for τ 0 =−τ 0

vE(z=zsl )=−τ 0

(1−i)2 fK

Volume transport in the IOBL

v dz−∞

0

∫ =−if

∂τ∂z

dz−∞

0

∫ =−ifτ 0

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Near the boundary in a neutrally stratified shear flow, the velocity profile is logarithmic, and is described empirically by:

Δu(d)u*0

=1κlog

dz0

wheρe d is τhe disτance fρom τhe boundaρy,Δv is τhe diffeρence in velociτy fρom τhe suρface τo d,z0 is τhe hydρaulic ρoughness of τhe suρface,and inτeρface friction velocity is defined by

u*0u*0 =τ 0

The Surface Layer

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By dimensional analysis or analogy with molecular viscosity,

eddy viscosity is the product of a scale velocity uτ and a m ixing lengτh, l, K =uτlIn τhe suρface layeρ, sτρess is appρoxim aτely consτanτ, so τhe consτiτuτive law is:

τ =K∂v∂z

=Ku*0

κz≈u*0u*0 ⇒ K =u*0κ z

so in τhe suρface layeρ, uτ =u*0 , l =κ z

The Surface Layer (continued)

A workable definition for the surface layer is the region

near the boundary where l varies linearly with distance

from the boundary.

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Seek by suitable choices for scales, to reduce a whole class of fluid dynamical regimes (in this case, neutrally stratified, rotating planetary boundary layers) to a common set of equations, with one solution.

Similarity

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dimensionless stress: Τ =τ / (u*0u*0 )

dim ensionless velociτy: U =v / u*0

dim ensionless veρτical cooρdinaτe: ζ =fz /u*0

ifv =∂τ∂z

⇒ iU =∂Τ∂ζ

Τ =fKu*0

∂U∂ζ

=K*Uζζ

com bine foρ a 2nd oρdeρ hom ogeneous eθn:

Τζζ −i

K*

Τ =0

wiτh b.c.'s: Τ(ζ → −∞)=0; Τ(ζ =0)=τ 0

u*0u*0

=1

Similarity (continued)

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Solution for the dimensionless stress:

Τ=edζ

wheρe d = i / K* =1+ i2K*

τ = u'w' + i v'w'

Dimensionless Ekman velocity

iU =∂Τ∂ζ

=1

dΤ

UE =vE

u*0

=−i

d=

12K*

(1−i)

Similarity (continued)

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For similarity, K* is constant in the Ekman layer,

and if Λ increases linearly in the surface

layer (where l =κ z ), the surface layer extent is

zsl=Λ*

κu*0 / f

K*=

flmax

u*0

=Λ*

ζ = fz / u*0

Surface Layer

Ekman Layer

Λ =κ z / (u*0

/ f )

From the LOW, the change in velocity across the

surface layer is:

Δuu*0

=1κlog

zml

z0=1κ(log

u*0

fz0+ log

Λ*

κ)

Similarity (continued)

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So now we can put together an “inverse drag law:”

v0

u*0

= vEu*0

+Δuu*0

=1κ

logu*0

fz0+ log

Λ*

κ+

12K*

(1−i)⎡

⎣⎢⎢

⎤

⎦⎥⎥

=1κ

logRo* −Am iB[ ]

Ro* =u*0 / ( fz0 ) is τhe suρface fρicτion Rossby no.A and B aρe sim ilaρiτy consτanτs (neuτρal sτρaτificaτion)

Similarity (continued)

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A =−logΛ*

κ−

κ2Λ*

+1+Λ*

2κ 2 ≈2.3

B=−κ2Λ*

+Λ*

2κ 2 ≈2.0

foρ Λ* =0.028

Γ

cw

β

Γ =v0u*0

=1κ

logRo* −A−iB[ ]

cw =u*02

v02 =Γ−2

Similarity (continued)

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•Rapid Melting

•Shallow Mixed Layers

•Internal Waves

Turbulence Length Scales:

Surface layer: κ z Neutral Ekman layer: Λ*u*0 / f

With buoyancy:

Buoyancy flux: w 'b ' =gρ

w'ρ '

By dimensional analysis: L = u*3

κ w'b'

Impact of Stratification

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lmax → Λ*u*0 / f for L0 = u*03 / (κ w 'b '

0→ ∞

λmax → Rcκ L0 for L0 → 0+

Rc is the critical flux Richardson number ≈ 0.2

Stratification (continued)

Extend the similarity theory to include buoyancy flux from rapid melting at the interface

With these asymptotes, a new stability parameter emerges

from the stipulation that K* is constant in the Ekman layer:

h• = 1+Λ*m*

κRc

⎛

⎝⎜⎞

⎠⎟

−1/2

m* =u*0 / ( fL0 ) is τhe ρaτio of τhe planeτaρy τo Obuκhov scales

h• ≤1 foρ neuτρal oρ m elτing condiτions.

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By analog with the neutral Rossby similarity development, formulate an expression for the dimensionless surface velocity:

McPhee, M. G, 1981: An analytic similarity theory for the planetary boundary layer stabilized by surface buoyancy, Boundary-Layer Meterol., 21, 325-339.McPhee, M. G., 2008: Air-Ice-Ocean Interaction: Turbulent Ocean Boundary Layer Exchange Processes, Springer, ISBN 978-0-387-78334-5.

U0=h•v0 / u*0Stratification (continued)

Γ(Ro* , m* )=1κ

logRo* −A(m* )−iB(m* )[ ]

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The impact of rapid melting is to reduce the turbulent length scale, and increase the velocity scale. It increases both A and B, which reduces the effective drag and increases the turning angle.

Example for:

u*0 =0.01 m s−1

z0 =0.05 mand m elτ ρaτe ρanging fρom 0 τo 30 cm /day

The rapidly melting ice (water about 2oC) drifts about 6.5 km farther in a day.

Stratification (continued)

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Stratification: Shallow Pycnoclines

In the western Arctic, there has been a remarkable increase in freshwater content over the past decade. This tends to stratify the water column closer to the surface, so the depth of the mixed layer adds a 4th length scale to the IOBL turbulence.

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To address this part of the problem requires treatment of fluxes in the upper part of the pycnocline, which are not amenable to the simple similarity approach.

However, a first-order turbulent closure model I call steady local turbulence closure (SLTC) uses the same similarity principles.

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When stress is mild, the impact of a shallow (20 m) mixed layer on surface velocity and drag is minor compared with one 2½ times as deep.

At higher stress, the difference is more apparent, manifested mainly as increased turning angle

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The SLTC model can also be used to examine a perennial problem with measurements made from drifting ice: how to scale results up to represent an entire floe or grid area in a numerical model.

During the ISPOL project in the western Weddell (2004-2005) we measured acoustic Doppler profiler currents in the upper ocean. I took every 3-h average, divided by the complex velocity at 30 m and then averaged the results for each depth for most of the project.

Large-scale Hydraulic Roughness

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Then model each 3-h segment using the measured T/S structure from the ship and matching the 20 m current. Each modeled profile is again nondimensionalized by the 30 m current, and averaged

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Location z0 Type

ISPOL (western Weddell)

40 mm Multiyear pack ice McPhee, Deep-Sea Res., 2008, doi;10.1016/j.dsr1012.2007

SHEBA (western Weddell)

49 mm Multiyear pack ice McPhee, Air-Ice-Ocean Interaction, 2008

NPEO (North Pole) 90 mm Multiyear pack ice, highly deformed

Shaw et al., JGR, 2008, doi: 10.1029/2007JC004550

MaudNESS (eastern Weddell)

4 mm Thin, first year ice Sirevaag et al., JGR, 2010, doi: 10.1029/2008JC005141

SLTC Roughness Estimates

Under fast ice in fjords and land-locked channels,

z0 is often very small, and can be considered

hydraulically smooth:

z0s ≈(n /u*0 )e−2

Τypical values: 3 τo 5 ×10−2 m m

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Other Roughness Estimates

McPhee, M.G., 1990: Small scale processes, in: Polar Oceanography, ed: W. Smith, Academic Press, 287-334.

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Recommendations for Drag Parameterization

• Recognize that drag coefficient and turning angle (β) depend on v0. Especially for thick ice, this can change the effective drag by as much as a factor of 2 over a reasonably speed range. For deep mixed layers and slow melting use Rossby similarity.

Γ =

1κ

logRo* −Am iB[ ]

• If water is warm, allow for stratification effects, either by incorporating them into a good IOBL model, or by applying the modified Rossby similarity. This may increase ice divergence, both in MIZ’s and where low concentration insolation has been intense.

Γ(Ro* , m* )=1κ

logRo* −A(m* )−iB(m* )[ ]

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Recommendations for Drag Parameterization

• If the pycnocline is shallow, recognize that it may appreciably change β in the opposite sense from Ro* similarity; e.g., expect larger β at higher speeds. There is less impact on drag magnitude.

47 m pycnocline

20 m pycnocline

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Recommendations for Drag Parameterization

• Regarding z0.

‣ For “normal” multiyear pack ice: 40-60 mm is a good guess.

‣ In highly deformed ice and MIZ’s: 60-120 mm.

‣ For drifting first year ice, with minor deformation: 1-4 mm.

‣ For undeformed fast ice: hydraulically smooth, unless there is platelet growth.

‣ Mixed first-year/multiyear: Use weighted average of the logarithms of z0 for the first and multiyear fractions.

• Internal Waves: These were observed to affect momentum flux during the 1984 MIZEX project. Details and a suggested parameterization are given by:

McPhee, M.G., and L.H. Kantha, 1989: Generation of internal waves by sea ice, J. Geophys. Res., 94, 3287-3302, 1989.