Ice/Ocean InteractionPart 1- The Ice/Ocean Interface

1. Background, rationale

2. Enthalpy and salt balance at the interface

3. The “Three-Equation” Interface Solution

4. Heat Flux Measurements and the Stanton Number for Sea Ice

5. Double-Diffusion and Melting

6. False Bottoms

7. Freezing: Is Double Diffusion Pertinent?

8. Summary

Basal (Ocean) Heat Flux

• Relatively small changes in open water fraction when sun angle is high are a major source of variability in the total surface heat balance.

• It is a misconception that mixed layers are like ice baths maintained at freezing temperature– in summer the Arctic mixed layer is typically several tenths of a degree above freezing, even in high ice concentrations. Under seasonal ice of the Weddell Sea, the mixed layer is nearly always at least 8-10 mK above freezing.

• In the perennial ice pack of the Arctic, transfer of heat from the ocean to the ice occurs mainly in summer, and in aggregate is as important in the overall heat balance as either the net radiative flux, or sensible and latent heat flux at the upper surface.

Day of 2002

NPEO JCAD/NPS BUOY 2002

McPhee et al., 2003, Geophys. Res. Lett.

dzdT

Ice

water

Thermal Balance at the Ice/Ocean Interface

w=w0+wp

Turbulent heat flux from ocean

Thermal conduction into ice

T0, S0 Latent heat source or sink

0ice

w

w h

Heat Equation at the Ice/Ocean Interface

• Heat conduction through the ice

• Latent exchange at the interface

• Turbulent heat flux from (or to) the ocean

00' ' 0Lq w T w Q

Heat conduction through the ice

ice pice

fresh ice

-1 -1 -1fresh

-1 -1 -1 -1 o

ice

/ thermal conductivity of sea ice

2.04 J m K s

0.117 J m K s psu C

is ice salinity in psu (pr

i i z

i

dTH K K T c q

dz

K K S T

K

S

-3

p

o -1 -1p

actical salinity scale)

is water density kg m

is specific heat of seawater, weak function of temperature

and salinity: ( 1.64 C, 30 psu) 4019 J kg K

is th

c

c

q

-1e "kinematic" form of the ice heat flux with units K m s

Latent heat exchange at the interface

(K) units erature with tempheat,

specificby divided volumebrineby adjustedfusion ofheat latent is Q

kg kJ 333.5 ice,fresh for fusion ofheat latent theis

)03.01(

L

1-fresh

0icefresh0Lp0iceL

L

wSLwQcwLH

Turbulent ocean heat flux

T p 0' 'H c w T

w=w0+wp

Turbulent salt flux from ocean

Advection into control volume

Advection out of control volume

Ice

Salt Balance at the Ice/Ocean Interface

S0

Sice

0' 'w S

Interface Salt Conservation Equation

0'' 0ice0 SSwSw

Parameterize turbulent flux of a quantity as proportional to the product of a scale velocity and and change in the quantity from the boundary to some reference level.

0 *0 *0mu u*0τ = u U

1

*0

0

1logm D

u dc

U z

Multiyear pack ice: z0 = 5 cm, d = 2 m (surface layer).

0.11m

Assume analogous expressions for scalar fluxes:

*00

*00

' '

' '

h

S

w T u T

w S u S

For multiyear ice, the corresponding melt rate would be about 1.3 m per day. During MIZEX 84 in the Fram Strait marginal ice zone we experienced melt rates of ~ 7 cm per day in conditions like these.

If by Reynolds analogy 0.11h m

Suppose u*0 = 0.01 m s-1, and T = 1 K.

Near the end of MIZEX we crossed an ocean front as the ice drifted south in response to a small storm. The mixed layer temperature rose abruptly, with rapid increase in basal melting.

McPhee, Maykut, Morison, J. Geophys. Res., 1987

On day 191, the average T > 1 K, and melt rate was around 7 cm per day.

*0 00' ' ( , , , , )Tw T F T u z

0 *0 0*

*0

' ', Pr,Reh h

T

w T u z

u T

If Reynolds analogy doesn’t work, resort to dimensional analysis

By the Pi Theorem five governing parameters, three with independent dimensions:

0 *0 0*

*0

' ', Sc,ReS h

S

w S u z

u S

Re Prp nSt

The Blasius solution for the laminar momentum and scalar contaminant boundary layers:

Where the Stanton number is the kinematic heat flux divided by the product of the far-field velocity and T. Exponents are p =1/2 and n = 2/3.

Owen and Thomson (1963) and Yaglom and Kader (1974) investigated heat and species concentration fluxes in laboratory flows over rough surfaces:

OT: p =0.45 and n = 0.8YK: p =1/2 and n = 2/3

*0 0 0

*0 0 0

0

0

h w L

S w ice

q u T T Q w

u S S w S S

The “three-equation” interface model:

Prescribe *0 , , , , , w w ice pu T S q S w

Unknowns: 0 0 0, , T S w

Third equation: 0 0T mS

Combine into a quadratic formula for S0

20 0( ) ( ) 0

LH ice p H P ice L wmS T T mS T S T T S T S

“Temperature Scales:”

*0H w

h

qT T

u

“Sensible”:

L SL

h

QT

“Latent”:

“Percolation”: *0

L pp

h

Q wT

u

small

“Two equation” approach

Assumption: T=Tw-T0 can be approximated by the departureof mixed layer temperature from freezing, i.e., that interface temperature is approximately the mixed layer freezing temperature.

*0 * *00' ' [ ( )]h w f ww T u T St u T T S

T0 and S0 are difficult to measure directly, but suppose the following:

00

( )' '' ' [ ]f w

p iceL

T Sw T qw S w S

Q m

From McPhee, Kottmeier, Morison JPO 1999

Yaglom-Kader Re* dependence

When is small, the quadratic solution depends on the ratio of exchange coefficients

q

/h SR which is a measure of the strength of double diffusion.

R = 1: heat and salt exchange is similar. Salt will enter the control volume at a rate necessary to keep 0 f wT T S

R >> 1: salt flux is inhibited relative to heat and the thermal driving is reduced. Melting is rate limited by double-diffusive effects in the transition sublayers.

0 f wT T S

/ /n

h S T SR 7 10 2 11.39 10 ; 6.8 10 mT S s

2 / 3 0.8 35 70n R

Thought experiment: -1*00; 0.015 m s ; 0.1 Kq u T

Using St* = 0.0057 (SHEBA average), the bulk formula basal heat flux is 34 W m-2. For R =1 and R = 70, find the h values that provide the same 34 W m-2 heat flux.

1 0.0058 0.098

70 70 0.0144 0.040h S

h S

R T

R T

*/ /h St T T

Now repeat the exercise using -1*00; 0.015 m s ; 2 Kq u T

keeping the same values for h and S as for the modest thermal forcing.

-2

-2

1 0.0058 681 W m

70 70 0.0144 988 W m

h S basal

h S basal

R H

R H

Respective melt rates are about 22 and 32 cm per day

false bottom “true” bottom

“water table”

The False Bottom ProblemNotz et al., J. Geophys. Res., 2003.

During the 1975 AIDJEX Project in the Beaufort Gyre, Arne Hansonmaintained an array of depth gauges at the main station Big Bear. Hereare examples showing a decrease in ice thickness for thick ice, but an increase at several gauges in initially thin ice.

Thick ice (BB-4 – BB-6) ablated 30-40 cm by the end of melt season. “Falsebottom” gauges showed very little overall ablation during the summer. The box indicates a 10-day period beginning in late July, when false bottoms apparently formed at several sites.

Estimated frictionvelocity for differentvalues of bottom surface roughness,z0 = 0.6 and 6 cm respectively

Changes in icebottom elevationrelative to a referencelevel on day 190, atthe “false bottom” sites.

Note that false bottoms appear to form at all sites during the relative calmstarting about day 205, and start migrating upward on or near day 210, whenthe wind picks up

.

Fresh W a ter Layer

Tu p =0 o C

Tw = -1.7 o C

h

Sea water ~ slightly abo ve freezing

Multiyear Ice

False Bottom

T 0 S 0 w T 0 w S 0 u *0

0Lower Boundary: ( / )

Upper Boundary:

bot w ice

up

h w

h

Assumptions:

•The upper water layer and the false bottom ice layer are fresh•The fresh water layer temperature is 0oC• The temperature gradient in the false bottom is linear.

0 0ice up ice

w p w p

K T T K mSq

c h c h

where h is the total thickness of the false bottom. The upper surface will then migrate upward at a rate

The kinematic heat flux in the false bottom ice is then downward:

wup

ice fresh

qh

Q

0( / )up w iceh h w

The time rate of change of thickness of the false bottom will then be

21 0 1 0

1*0

1 ( 1 ) 0Lw ice w ice L w

S LL

h

ice

w p h

mS T T mS S T S T S

QT

K

c u h

The quadratic equation for the lower interface is similar to earlier but modified for the different ice heat flux condition:

false bottom “true” bottomupward heat flux

down

“water table”

Aside from maybe helping us estimate h, are false bottoms of more than academic interest?

They are effective at shielding thin ice from ocean heat hence potentially important for ice-albedo feedback

1

They are like radiators pumping heat into the mixed layer rather than sucking it out.

2

Summary for melting

Transport of salt across the interface is much slower than heat, and effectively controls the melting rate

The exchange ratios for heat and salt (h and S) are difficult to measure, but are constrained by the bulk Stanton number, which is measurable

Collection of fresh water in irregularities in the ice undersurface both protects thin ice from melting and slows the overall heat transfer out of the mixed layer. This retards (provides a negative feedback to) the summer ice-albedo feedback

Double Diffusion during Freezing: A Mechanism for Supercooling or Frazil Ice Production?

If the same sort of double diffusion occurs in the upper ocean (transition sublayers) during freezing, then if heat leaves the upper ocean faster than salt enters, there is potential for supercooling, and if nucleation sites are available, for accumulation of frazil ice.

Impact of exchange coefficient ratio on freezing with 01.00* u

1Sh cc

1Sh cc

30Sh cc

30Sh cc

1/ Sh

30/ Sh

1/ Sh

30/ Sh

Straight congelation growth Growth with frazil accretion

SonTek ADVOcean (5 Mhz)

SBE 03 thermometerSBE 07 microstructureconductivity meter

SBE 04 conductivity meter

SonTek Instrument Cluster

We made measurements as part of the primary experiment from day 67 to 70, then during the UNIS student project a week later. Using the observation that the ice was hydraulically smooth we can estimate the stress from a current meter that recorded continuously at 10 m depth, giving us a 24-day record to provide the momentum forcing for a numerical model.

65 70 75 80 85 900

1

2

3

4

5x 10

-3

Day of 2001m

s-1

Run D060824a: u* = 1/2, 1 m from Ice

67.5 68 68.5 69 69.5 700

1

2

3

4

5x 10

-3

Day of 2001

m s

-1

Run D060824a: u* = 1/2, 1 m from Ice

Model

TIC

Model

TICStudent TIC

65 70 75 80 85 903

4

5

6

7

Day of 2001

mm

d-1

Ice Growth Rate

R= 1

R= 5

R=35

During the initial project, the temperature gradient near the base of the ice indicated a conductive heat flux of around 20 W m-2. We used this as a constant flux in the model, shown above for 3 different values of R. The dashed line is the mean growth rate determined by comparing the ice thickness measured at the start and end of the period. The second plot shows the modeled and observed salinity.

65 70 75 80 85 900

2

4

6

8

10

12

Day of 2001

W m

-2

Turbulent Heat Flux, 1 m from Ice

R=1

R=5

R=35TIC

Student TIC

66 66.5 67 67.5 68 68.5 69 69.5 700

1

2

3

4

Day of 2001

W m

-2

Turbulent Heat Flux, 1 m from Ice R=1

R=5

TIC

From the numerical model we can estimate the turbulent heat flux 1 m below the interface for the various R values, then compare the model output with measurements made during the two observation period. This provides strong evidence that the double-diffusive effect is very small (R = 1) when ice is freezing.

Conclusions

• During melting, double diffusion effects are paramount, and ice dissolves as much as it melts

• False bottoms (a) may protect thin ice from the impact of ocean heat flux during summer; (b) provide a means of determining the ratio of diffusivities appropriate for melting ice.

• During freezing, it appears that double diffusive tendencies are relieved near the interface by differential ice growth, so that supercooling and frazil production are limited during congelation growth