# One Eye is All You Need: Lightweight Ensembles for Gaze Estimation with
Single Encoders

Rishi Athavale
Vytal.ai
McLean, VA
<EMAIL_ADDRESS>
Lakshmi Sritan Motati
Vytal.ai, Harvard Medical School
McLean, VA
<EMAIL_ADDRESS>
Rohan Kalahasty ✉ †
Vytal.ai, Harvard Medical School
McLean, VA
<EMAIL_ADDRESS>

###### Abstract

Gaze estimation has grown rapidly in accuracy in recent years. However, these
models often fail to take advantage of different computer vision (CV)
algorithms and techniques (such as small ResNet and Inception networks and
ensemble models) that have been shown to improve results for other CV
problems. Additionally, most current gaze estimation models require the use of
either both eyes or an entire face, whereas real-world data may not always
have both eyes in high resolution. Thus, we propose a gaze estimation model
that implements the ResNet and Inception model architectures and makes
predictions using only one eye image. Furthermore, we propose an ensemble
calibration network that uses the predictions from several individual
architectures for subject-specific predictions. With the use of lightweight
architectures, we achieve high performance on the GazeCapture dataset with
very low model parameter counts. When using two eyes as input, we achieve a
prediction error of 1.591 cm on the test set without calibration and 1.439 cm
with an ensemble calibration model. With just one eye as input, we still
achieve an average prediction error of 2.312 cm on the test set without
calibration and 1.951 cm with an ensemble calibration model. We also notice
significantly lower errors on the right eye images in the test set, which
could be important in the design of future gaze estimation-based tools.

††$\dagger$ denotes corresponding author.

##  1 Introduction

In the past years, advances in gaze tracking technology have allowed for its
use in a large variety of fields. An abundance of research has shown that many
neurological diseases, including Parkinson’s, Alzheimer’s, Schizophrenia, and
Autism, have distinctive, early onset bio-markers that can be found through
eye movement analysis [6]. For example, studies have shown that people with
Alzheimer’s have higher correction rates during rapid eye movements and
fixations [7]. Gaze tracking headsets can be used to diagnose diseases based
on these bio-markers by analyzing saccadic changes. Gaze tracking is also
heavily utilized for advertising research, where it is used to see which parts
of the ad are focused on by users. However, the current state-of-the-art gaze
tracking devices cost thousands of dollars, preventing the widespread adoption
of the technology. Lower-cost gaze tracking methods need to be developed.

Current methods for gaze tracking leverage modern advancements in machine
learning (ML), computer vision (CV), and graphics processing units (GPUs) to
train ML models to make gaze predictions. While it is common for ML-based gaze
estimators to mainly make use of frontal images of a subject, many methods
have utilized less conventional forms of data such as video [8], 3D models
[4], and 3D facial reconstructions [9].

More traditional methods of ML-based gaze estimation, however, instead use an
image of a subject, with some using the full frontal face and some using
cropped eye images. In order to improve performance, they often calibrate a
model to a specific user by first collecting data from them and then tuning
the model to the user-specific data [5, 11]. This allows models to fit to the
specific features present in a specific user.

These methods, although somewhat simple, are generally less computationally
intensive than methods that use more complex data inputs. Furthermore, they
are also able to achieve impressive results by training on large amounts of
data. One of the largest gaze prediction datasets is the GazeCapture dataset,
which includes nearly 2.5 million frames taken from over 1450 people [5]. In
their paper "Eye Tracking for Everyone", Kafka et al. introduce not only the
GazeCapture dataset but also an ML architecture to generate gaze estimations.
Their method takes as input the face and eyes from a frame as well a face
grid. The face image, right eye image, and left eye image are each passed
through their own feed-forward convolutional neural network (CNN), which
extracts their features. The features of the face grid are extracted using a
feed-forward neural network (NN). The features for both of the eyes are then
passed through a fully connected (FC) layer to generate features that
represent both of the eyes. These feature vectors are then concatenated and
passed through a feed-forward NN which outputs a $(x,y)$ coordinate that
represents the gaze prediction. Without calibration, this method was able to
achieve a prediction error of 1.71 cm on data from mobile phones and a
prediction error of 2.53 on data from tablets [5].

Research from Valliappan et al. was able to achieve impressive results using
the GazeCapture dataset while only using eye images and eye landmarks. They
utilized a "dual tower" approach, in which each eye image (with the left eye
being flipped horizontally) has its features extracted by the same CNN. They
also used a feed-forward NN to extract features from eye landmarks, which
represent the top left and bottom right coordinates of each eye. The eye
features and landmark features are then concatenated together, after which
they are passed through a feed-forward NN which outputs a $(x,y)$ coordinate
that represents the gaze prediction. With this approach, Valliappan et al.
were able to achieve a prediction error of 1.92 cm on the GazeCapture dataset
[11].

Current gaze estimation methods generally have the advantage of being
lightweight while being able to leverage large datasets to achieve low
prediction errors. However, these methods fail to make use of other CV
techniques that could both improve results while keeping gaze estimators
lightweight. The ResNet and Inception architectures, for example, have
achieved great success on many CV tasks such as the ImageNet challenge (on
which ResNet achieved a top-5 error rate of 3.57% in 2015 [3] and Inception
achieved a top-5 error rate of 6.67% in 2014 [10]). Additionally, methods such
as ensemble models (which combine predictions from multiple models) have
helped improve results in CV tasks such as image classification [2]. The fact
that traditional gaze prediction methods don’t utilize such methods could be
limiting their performance.

Current gaze estimation methods also often rely on having an image of a
person’s face [5] or an image of both a person’s eyes [11]. This limits their
applicability because, in real-world conditions, it is not uncommon for a
person’s face or one of their eyes to be partially obscured, for an eye-
detection model to only successfully detect one eye, or even for a user to be
missing an eye. A more practical method of gaze estimation, therefore, would
be one that can make a prediction from just a single eye.

##  2 Methodology

To test the effectiveness of other computer vision methods, we modified the
CNN architecture in [11] to make three new versions: one with a ResNet-based
architecture, one with an Inception-based architecture, and one with an
InceptionResNet-based architecture. In addition, we made modified versions of
all of these models that would only take in one eye as input in order to test
one-eye gaze estimation. After training and evaluating these models based on
mean Euclidean distance (in centimeters), we calibrated them to testing set
subjects by fitting a lightweight support vector machine (SVM) to training set
frames for a specific subject.

###  2.1 Data Preprocessing

To train and evaluate models for gaze estimation, we used the GazeCapture
dataset. This dataset includes nearly 2.5 million frames along with important
metadata (such as the size of the screen and the location of the eyes and
face) for over 1450 subjects [5]. We use frames taken from iPhones in portrait
orientation and with valid eye detections, resulting in a dataset of 501,675
frames. Similar to [11], we also modified the metadata for each frame to
include automatically-extracted eye corner landmarks in order to provide the
location of the subject’s eyes in the frame. For each frame, the eye regions
(with each one being resized to 128x128) and the eye landmarks (coordinates of
the top left and bottom right corners of the eye regions) were used.

To split the dataset into training, validation, and testing sets, we used a
similar method to [11] in that we split our dataset based on unique ground
truth points. This means that while frames from the same subject can appear in
different datasets, frames corresponding to a particular ground truth point
will not appear in different datasets (thereby preventing data leakage). Using
an 80/8/12 dataset split, the training set had 399,093 frames, the validation
set had 43,558 frames, and the testing set had 59,024 frames.

###  2.2 Proposed Approach

Current gaze estimation methods have largely opted for basic CNN
architectures. However, more complex methods such as the ResNet and Inception
architectures have been shown to perform better than traditional CNNs in a
variety of tasks, such as the ImageNet challenge [3, 10]. In order to
integrate preexisting gaze estimation architectures with more complex CV
methods, we propose modified versions of the model architecture used in [11]
in which the eye-feature-extractor (which in [11] is a basic feed-forward CNN)
is replaced with a small ResNet architecture, a small Inception architecture,
or a small InceptionResNet architecture. By basing these models on the one
used by [11] and keeping the model architectures small, the proposed models
continue to be lightweight while being able to better map eye images to
features due to having more parameters.

A weakness of current gaze estimation methods is that they require either a
frontal view of a subject’s face or images of both of their eyes [5, 11]. This
limits the practical use of these models because it makes them less useful in
cases where a person’s face is partly obscured or when their head is turned. A
significantly more practical method would be one that could make a gaze
estimation from an image of a single eye. Not only would this allow for more
accurate gaze estimations in the aforementioned cases, but it would also be
usable for frames in which a given eye-detection model can identify only one
eye.

Current gaze calibration methods involve either the retraining of a full model
or fitting a lightweight model to map the activations from the base model to
the ground truth points of the subject [11]. The first method is extremely
impractical as not only does retraining a full model take significantly more
time than a lightweight calibration model, but it would also be difficult to
scale to a large user-base as each retrained model would have to be separately
saved. While this could be avoided by re-calibrating the model with every use,
this would be inconvenient and time-consuming for users. The second method is
also limited by the fact that the lightweight calibration model can only base
its predictions on features from the base model. This means that if there is a
problem with the base model’s features, it can be nearly impossible for the
calibration model to make accurate predictions for a subject. To improve on
the limitations of current calibration methods, we propose an ensemble
approach in which the lightweight calibration model is fit on features from
multiple base models. Ensemble models have proven to be successful on a
variety of CV tasks (such as the ImageNet challenge [2]) because of their
ability to utilize information from multiple models. By using an ensemble
approach, the lightweight calibration model will be less likely to be limited
by problems with base model features because if multiple base models are used,
it is more likely one of them will have features that can better fit the
subject.

###  2.3 Model Architectures

The proposed model architectures are modified versions of the one used in
[11]. While the landmark feature extractor (the feed-forward neural network
that processes the eye corner landmarks) is largely kept the same, the eye
image feature extractor (the CNN that processes the eye images) is modified.

####  2.3.1 Regular CNN architecture

The regular CNN architecture (which in results will be referred to as "CNN")
is based on the one presented in [11]. The architecture consists of two CNN
"towers" (which process the eye images) and a landmark model which processes
the eye landmarks with a feed-forward neural network. One CNN tower processes
the right eye image and the other processes the left eye image (which is
flipped horizontally). The two CNN towers are the same model and share
weights. Each tower consists of three convolutional layers with kernel sizes
of 7x7, 5x5, and 3x3, output channels of 32, 64, and 128, and strides of 2, 2,
and 1. Each convolutional layer is followed by an average pooling layer with a
pooling size of 2x2. The output is of size 2x2x128, which when flattened is of
size 512 [11].

The landmark model is a feed-forward neural network with three layers with
128, 16, and 16 units respectively. Each of these layers is followed by batch
normalization and a rectified linear unit (ReLU) layer [11].

The feature vectors from the eyes (each of size 512) are then concatenated
with the feature vector from the eye landmarks (which has a size of 16). The
resulting concatenated feature vector is then processed through a feed-forward
neural network with three layers with 8, 4, and 2 units respectively [11]. The
CNN architecture used is shown in Figure 1a.

(a)

(b)

(c)

(d)

Figure 1: Model architectures that we implement.

####  2.3.2 ResNet architecture

The ResNet architecture used includes only two residual blocks in order to
keep the model small and lightweight. It consists of three blocks, with each
consisting of two convolutional layers (with kernel sizes 4x4 for the first
block, 3x3 for the second block, and 2x2 for the third block), two batch
normalization layers (one after each convolutional layer), a leaky rectified
linear unit (ReLU) layer (after the first convolutional layer), an average
pooling layer with a pooling size of 2x2 (after the second convolutional
layer) and a dropout layer (after the second convolutional layer). The second
convolutional layer in each block had a stride of 2. The output channels for
each block were 32, 64, and 128 respectively. In order to create residual
blocks, the first and second blocks are added as are the second and third
blocks. To do this, a down-sampling block is applied to the first block so it
will be the same size as the second block, and another down-sampling block is
applied to the second block so it will be the same size as the third block.
The first down-sampling block has 64 output channels and consists of a
convolutional layer with a kernel size of 5x5 and a stride of 2, a batch
normalization layer, an average pooling layer with a pooling size of 2x2, and
a dropout layer. The second down-sampling block is the same except it has 128
output channels and its convolutional layer has a kernel size of 3x3 and a
stride of 1. The resulting output is of size 2x2x128 (which when flattened is
of size 512). The diagram for the ResNet architecture used is shown in Figure
1b.

####  2.3.3 Inception architecture

The Inception architecture, like the ResNet architecture, consists of three
blocks, two of which are inception blocks. The first block consists of a
convolutional layer with a kernel size of 7x7, a stride of 2, and 32 output
channels, a batch normalization layer, a leaky ReLU layer, an average pooling
layer with a pooling size of 2x2, and a dropout layer. The second block
consists of four branches that are concatenated together: one with one
convolutional layer with a kernel size of 1x1, a stride of 1, and 32 output
channels and a batch normalization layer, one with two convolutional layers
(with kernel sizes of 1x1 and 3x3 respectively and each having a stride of 1
and 32 output channels) and a batch normalization layer, one with two
convolutional layers (with the first one having a kernel size of 1x1 and the
second having a kernel size of 5x5 and a padding value of 1, and each having a
stride of 1 and 8 output channels) and a batch normalization layer, and one
with a max pooling layer with a pooling size of 3x3, a stride of 1, and
padding value of 1, a convolutional layer with a kernel size of 1x1, a stride
of 1, and 8 output channels, and a batch normalization layer. After the four
branches are concatenated, they are passed to a convolutional layer with a
kernel size of 5x5, a stride of 2, and 64 output channels, a batch
normalization layer, a leaky ReLU layer, an average pooling layer with a
pooling size of 2x2, and a dropout layer. The four branches, concatenation,
and subsequent layers are part of the second block. The outputs from the
second block are then passed to the third block, which consists of four
branches. These branches are the same as the ones in the second block, except
that the number of output channels is doubled. The four branches, after being
concatenated, are passed to a convolutional layer with a kernel size of 3x3, a
stride of 1, and 128 output channels, a batch normalization layer, a leaky
ReLU layer, an average pooling layer with a pooling size of 2x2, and a dropout
layer. The resulting output is of size 2x2x128 (which when flattened is of
size 512). The diagram for the Inception architecture used is shown in Figure
1c.

####  2.3.4 InceptionResNet architecture

The InceptionResNet architecture integrates the aforementioned ResNet and
Inception architectures. Its architecture is the same as the Inception
architecture described above, but with residuals that are obtained by down-
sampling the first and second blocks and added to the second and third blocks
respectively. Since the shapes of the outputs of each block are the same, the
down-sampling blocks are the same as the ones described in the ResNet
architecture. The diagram for the InceptionResNet architecture used is shown
in Figure 1d.

####  2.3.5 One-Eye Gaze Estimation

The one-eye gaze estimation models use the same eye image feature extraction
architectures (the CNNs that process the eye images) as those used in the two-
eye gaze estimation models. However, they will only process one eye image (as
opposed to the two-eye gaze models, which use a "dual tower" approach to
process both eye images). When left-eye images are passed to the one-eye gaze
estimators, they are not flipped horizontally. This is because flipping the
left eye image horizontally when only one eye image is given would make it
harder for the model to determine what direction the eye is looking in. For
example, if a left eye that is looking to the left is given to the model and
it is flipped horizontally, it would be difficult for the model to tell if it
was a left eye looking leftward or a right eye looking rightward. The landmark
feature extractor (the feed-forward neural network that processes the eye-
corner landmarks) is largely the same, except that the input shape is of size
4 due to only processing 2 coordinates.

During training, each frame is randomly assigned a 0 or a 1. For frames that
are assigned a 0, their right eye image and right eye landmarks will be used,
and for frames that are assigned a 1, their left eye image and left eye
landmarks will be used.

###  2.4 Ensemble-based Calibration

Since data was split based on ground truth point rather than subject, each
subject has a number of frames in the training and the test set. For each
frame of a subject present in the training set, each model’s (regular CNN,
ResNet, Inception, and InceptionResNet) activations before the linear output
layer are collected and concatenated into a feature vector. This results in a
set of feature vectors that each correspond to a ground truth value. A support
vector machine (SVM) is then trained to map the feature vectors to their
respective ground truth point. After the SVM is trained, the feature vectors
are then obtained for the subject’s frames present in the test set. The SVM’s
predictions are then compared with the ground truth points using the mean
Euclidean distance in centimeters. This process is done for every subject in
the dataset to determine the average performance of the calibration method
across the dataset.

For the one eye models, calibration is tested in three different ways. First,
the process described above is performed but only with data for the subjects’
right eyes (meaning that feature vectors are obtained by passing the right eye
image and right eye landmarks corresponding to a frame). This is then done
again but with data for the subjects’ left eyes. For the final method, both
eyes are used. This is done by getting the feature vectors for the right eye
data and the feature vectors for the left eye data and concatenating them
together. This concatenated feature vector is then used as the input to the
SVM, which is trained to map the concatenated feature vectors to the ground
truth points. The diagram for the ensemble-based calibration is shown in
Figure 2.

Figure 2: Diagram for ensemble-based calibration.

###  2.5 Training

The models were all trained with the Adam optimizer, an initial learning rate
of 0.016, an exponential learning rate scheduler, and a batch size of 256. The
training loss was the mean squared error between the predicted coordinates and
the true coordinates. Mean squared error was also used to evaluate the model’s
performance on the validation set (with a lower mean squared error indicating
a better model and a higher mean squared error indicating a worse model).
Training was done for 50 epochs, and the weights with which it performed best
on the validation set (obtained the lowest validation mean squared error) were
saved.

###  2.6 Evaluation

To evaluate the quality of a gaze prediction, the Euclidean distance (in
centimeters) between the predicted coordinate $(\hat{x},\hat{y})$ and the true
coordinate $(x,y)$ is obtained, and it is calculated using the following
formula:

$\sqrt{(\hat{x}-x)^{2}+(\hat{y}-y)^{2}}$

To evaluate a gaze estimator’s performance on a dataset, the mean of the
Euclidean distances for every frame is calculated.

###  2.7 Implementation Details

The developed models were trained and evaluated on the GazeCapture dataset,
which was split into train, validation, and test sets. The models were made
using the PyTorch framework. The training was done with an AWS p3.2xlarge
instance, which included a Tesla V100 GPU with 16 GB of memory. Our code can
be found here: GitHub.

##  3 Experiments and Results

Table 1: Mean Euclidean distance (in centimeters) for each model with two eyes as input on the validation and testing sets. Model | Validation Mean Error (cm) | Test Mean Error (cm)
---|---|---
CNN | 1.694 | 1.692
ResNet | 1.604 | 1.607
Inception | 1.590 | 1.591
InceptionResNet | 1.683 | 1.683
Calibration | N/A | 1.439
Table 2: Mean Euclidean distance (in centimeters) for each model with one eye as input on the validation and testing sets. Model | Eye Data | Validation Mean Error (cm) | Test Mean Error (cm)
---|---|---|---
CNN | Right Eye | 2.073 | 2.078
ResNet | Right Eye | 2.000 | 2.013
Inception | Right Eye | 1.934 | 1.947
InceptionResNet | Right Eye | 2.042 | 2.049
CNN | Left Eye | 2.647 | 2.685
ResNet | Left Eye | 2.605 | 2.611
Inception | Left Eye | 2.678 | 2.708
InceptionResNet | Left Eye | 2.651 | 2.691
Calibration | Right Eye | N/A | 1.800
Calibration | Left Eye | N/A | 2.102
Calibration | Both Right and Left Eye | N/A | 1.774
Table 3: Average of left-eye and right-eye mean Euclidean distances (in centimeters) for the one eye models on the validation and testing sets. Model | Average Validation Mean Error (cm) | Average Test Mean Error (cm)
---|---|---
CNN | 2.360 | 2.382
ResNet | 2.303 | 2.312
Inception | 2.306 | 2.327
InceptionResNet | 2.346 | 2.370
Calibration | N/A | 1.951

The four different models (CNN, ResNet, Inception, and InceptionResNet) were
trained on the training set for 50 epochs whilst saving the model weights that
achieved the lowest validation loss. This was done for both the two-eye and
one-eye models. Figure 3 shows the training and validation curves for each
model. As shown in Table 1, the Inception model did the best on the validation
and test sets out of the four base models, with it achieving a validation mean
error of 1.590 cm and a test mean error of 1.591 cm. Furthermore, the regular
CNN model did the worst out of the four. The low variability between the
validation and the test mean errors show that the models likely did not
overfit.

(a)

(b)

(c)

(d)

Figure 3: Graphs of the training or validation loss over training for the two
eye and one eye models.

As shown in Table 2, the Inception model achieved the lowest errors on the
validation and test sets when given only right eye data (1.934 cm on the
validation set and 1.947 cm on the test set) and the ResNet model achieved the
lowest errors on the validation and test sets when given only left eye data
(2.605 cm on the validation set and 2.611 cm on the test set). One-eye models
tended to perform significantly better when given right eye data as opposed to
left eye data. As shown in 3, the ResNet model performs the best on average
out of the four base models, with its average error being 2.303 cm on the
validation set and 2.312 cm on the test set.

For both the two-eye and one-eye models, ensemble calibration with a
lightweight SVM was able to dramatically reduce error. As shown in Table 1,
the two-eye calibration model achieved a test mean error of 1.439 cm (0.152 cm
less than Inception’s test mean error). The one-eye calibration model, as
shown in Table 2, achieved a right eye mean error of 1.800 cm and a left eye
mean error of 2.102 cm, the average of which (as shown in Table 3) is 1.951 cm
(0.361 cm less than the average error for ResNet on the test set).
Furthermore, the one-eye calibration model was able to achieve a test mean
error of 1.774 cm when the SVM was given concatenated feature vectors from
both left and right eye data, as shown in Table 2.

##  4 Conclusion

The results of this study show the effectiveness of the ResNet and Inception
architectures as well as ensemble calibration for gaze tracking. We were able
to show that ResNet, Inception, and InceptionResNet models were able to
significantly outperform standard CNN architectures, which are commonly used
for gaze estimation. Our Inception base model, which achieved a test mean
error of 1.591 cm, was also able to outperform the base models created by [5]
(which achieved an error of 1.71 cm) and [11] (which achieved an error of 1.92
cm) for GazeCapture. We were also able to show how small versions of the
ResNet, Inception, and InceptionResNet architectures could make accurate gaze
predictions while remaining lightweight.

Additionally, we were able to show how gaze estimators can still be fairly
accurate even when only given data for a single eye. Despite the one-eye gaze
models having half as much data per example as the two-eye gaze models, they
were able to achieve an average error of 2.312 cm without calibration (based
on ResNet performance) and 1.951 cm with calibration. When given data for both
eyes, the calibration model performs even better, achieving an error of 1.774
cm. Not only does this show the high accuracy of one-eye gaze estimation, but
it also shows that the one-eye gaze models were able to decipher unique and
useful features for left and right eyes (as otherwise, the one-eye calibration
model would not have seen a significant decrease in test mean error when using
data from both eyes).

Further research should seek to implement the calibration methodology outlined
in this paper with real-world video data. This would be done by recruiting
participants to record 30-60 seconds on video where their gaze coordinates
will be tracked (likely by instructing them to look at a specific point). This
data can then be used to calibrate the calibration model. This would likely
achieve better calibration results than the ones obtained in this paper, as we
were limited to using data from subjects already in the GazeCapture dataset.
Having video data where a participant is instructed to look at a specific
point can be used to get a wider variety of frames and ground truth points for
a participant. It would also be more accurate to how gaze calibration would be
implemented in the real world. Other research should also seek to test other
CV model architectures such as EfficientNet and MobileNet. Lightweight
versions of these models would have to be created (similar to how lightweight
versions of ResNet, Inception, and InceptionResNet were created for this
study). Additionally, future research should seek to pretrain the CNN models
on the ImageNet database and then retrain them on GazeCapture in order to see
if transfer learning would improve results by allowing the models to start
with an understanding of a variety of features before training. Pretraining
models on ImageNet has been used for other gaze tracking models, such as L2CS-
Net, to achieve impressive results [1].

The results of this study have important implications for gaze tracking with
machine learning. They not only show that future research should focus on more
complex CNN architectures such as ResNet and Inception instead of just regular
CNNs, but they also show how one-eye gaze tracking can be a viable method for
gaze prediction. This would allow impaired individuals with only a single eye
to use gaze tracking software. It also has the potential to make gaze tracking
software more applicable to real-world scenarios. In real-world data, it is
common to have images where people aren’t looking straight at a camera, one of
their eyes is blocked by long hair, the eye detection software only detects a
single eye for a given frame, and more cases where gaze predictions would have
to be made based on a single eye. The findings of this research and the models
produced have the potential to make gaze tracking more applicable to
e-commerce, digital accessibility, and medical diagnosis.

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# Universal scaling regimes in rotating fluid turbulence

Abhik Basu  abhik.basu<EMAIL_ADDRESS>Theory Division, Saha
Institute of Nuclear Physics, Calcutta 700064, West Bengal, India  Jayanta K
Bhattacharjee<EMAIL_ADDRESS>Department of Theoretical
Physics, Indian Association for the Cultivation of Science, 2A and 2B Raja S C
Mullick Road, Calcutta 700032, West Bengal, India

###### Abstract

We analyse the scaling properties of the energy spectra in fully developed
incompressible turbulence in forced, rotating fluids in three dimensions (3D),
which are believed to be characterised by universal scaling exponents in the
inertial range. To elucidate the scaling regimes, we set up a scaling analysis
of the 3D Navier-Stokes equation for a rotating fluid that is driven by large-
scale external forces. We use scaling arguments to extract the scaling
exponents, which characterise the different scaling regimes of the energy
spectra. We speculate on the intriguing possibility of two-dimensionalisation
of 3D rotating turbulence within our scaling theory. Our results can be tested
in large scale simulations and relevant laboratory-based experiments.

##  I Introduction

Nonequilibrium systems are described by the appropriate equations of motion
for the relevant dynamical variables and exhibit much richer universal
behavior than usually observed in equilibrium critical dynamics hal .
Hydrodynamic turbulence in fluids, described by the Navier-Stokes equation
land ; frish  for the evolution of the velocity field $\bf v$, is a prime
example of an out of equilibrium system, due to the external drive acting on
the fluids. Interestingly, fully developed fluid turbulence in three- (3D) and
two- (2D) dimensions show markedly different behavior: In 3D, the energy
spectra follow the well-known K41 result for homogeneous and isotropic 3D
hydrodynamics turbulence where the one-dimensional energy spectra $E(k)\sim
k^{-5/3}$ (hereafter K41) in the inertial range, where $k$ is a wavevector k41
. This K41 result is quite robust and universal, and found in wide-ranging
natural systems, e.g., shear flows shear , viscoelastic fluids visco  and jet
flows jet . In contrast, 2D turbulence is characterised by an inverse cascade
of energy at very large scales with $E(k)\sim k^{-5/3}$, and forward cascade
of enstrophy with $E(k)\sim k^{-3}$ at intermediate scales 2d1 ; 2d2 ; 2d3 ;
2d4 ; 2d5 .

Rotating turbulence, i.e., turbulence in a rotating fluid, is a naturally
occurring phenomenon in many astrophysical and geophysical flows, as well as
in laboratory-based engineering fluid flows. The presence of the Coriolis
forces is the distinctive feature of rotating turbulence, which should affect
the large-scale scaling properties of rotating turbulence. In spite of
extensive studies, there is still no good agreement on the scaling of the
energy spectra in rotating turbulence, in particular how the Coriolis forces
affect the scaling of the spectra at very large scales. Varieties of
analytical, numerical, or experimental investigations of either the forced or
the decaying rotating turbulent fluid suggest that the kinetic energy spectra
in the rotation-dominated small-$k$ regions should scale as $E(k)\sim
k^{-m},\,m\in(2,3)$ 2ds1 ; 2ds2 ; 2ds3 ; 2ds4 ; 2ds5 ; 2ds6 ; 2ds7 ; 2ds8 ;
2ds9 ; 2ds10 ; 2ds11 ; 2ds12 ; 2ds13 ; 2ds14 ; 2ds15 . Recent perturbative
studies indicate that the one-dimensional kinetic energy spectra made out of
the velocity component parallel to the rotation axis scales as $k^{-5/3}$,
indistinguishable from the K41 prediction. In contrast, the one-dimensional
kinetic energy spectra made out of the velocity components lying in a plane
normal to the rotation axis scales as $k^{-3}$, different from the K41 scaling
ab-jkb-rot . The precise forms of the scaling of the energy spectra in
rotating turbulence however is still not well-settled.

There is a degree of formal similarity between the (linearised) equations of
motion of rotating turbulence, which is nothing but the Navier-Stokes equation
in a rotating frame (see below) and the equations for magnetohydrodynamic
turbulence (MHD) in the presence of a mean magnetic field $B_{0}$ jackson ;
arnab . In the former case, the Coriolis forces lead to oscillatory modes,
whereas in the MHD case, a non-zero $B_{0}$ gives rise to propagating Alfv́en
waves. Strong Alfv́en waves are known to make the energy spectra in MHD
anisotropic, and change the scaling as well jkb-mhd . In the same vein, strong
Coriolis forces should make the scaling of energy spectra in rotating
turbulence anisotropic and also different from its isotropic counterpart
(i.e., the K41 scaling).

In this work, we revisit the universal scaling of energy spectra in forced,
statistically steady rotating turbulence in its inertial range. To this end,
we have set up a scaling theory to study the scaling of the energy spectra in
the inertial range. We cover both the weak and strong rotation limits. In the
former case, unsurprisingly, the K41 result is found. With stronger rotation,
anisotropic scaling with different exponents ensues. In particular, in the
wavevector region $k_{\perp}\gg k_{\parallel}$, our scaling theory gives the
scaling of the 2D spectra $E(k_{\perp},k_{\parallel})$, where ${\bf
k}_{\perp}$ and $k_{\parallel}$ are the components of the wavevector $\bf k$
in the plane perpendicular to the rotation axis (here $\hat{z}$-axis) and
along the rotation axis, respectively. We find $E(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-5/2}k_{\parallel}^{-1/2}$ for $k_{\perp}\gg k_{\parallel}$, which
agrees with the Kuznetsov–Zakharov–Kolmogorov spectra predicted by the weak
inertial-wave turbulence theory for the rotating fluids 2ds7 . We show that
this result is unaffected by nonlinear fluctuation corrections at the one-loop
order. We further demonstrate that this result could be obtained by demanding
that the cascade of the kinetic energy flux is hindered by a non-zero
helicity, which is naturally present in a rotating fluid. In the opposite
limit of $k_{\perp}\ll k_{\parallel}$, we get $E(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-1}k_{\parallel}^{-2}$. We also show perturbatively in the rotation
$\Omega$ that the kinetic energy flux is indeed reduced by it. The remainder
of the article is organised as follows. In Sec. II, we set up the forced
Navier-Stokes equation in a rotating fluid. Then in Sec. III we set up the
scaling arguments. Then next in Sec. III.1, we revisit the K41 scaling scaling
in an isotropic, nonrotating fluid turbulence, and show how are scaling theory
reproduces it. Next, in Sec. III.2, we show that for weak rotation, the energy
spectra again show the K41 scaling. Then in Sec. III.3 we study the 2D
anisotropic energy spectra in the opposite limit of large $\Omega$. In Sec. IV
we discuss and summarise our results. We provide some technical results,
including a perturbative demostration of the reduction of the kinetic energy
flux by helicity, in the Appendix for interested readers.

##  II Turbulence in a rotating fluid

The Navier-Stokes equation for the velocity field ${\bf v}({\bf r},t)$ in a
rotating frame with rotation ${\boldsymbol{\Omega}}=\omega\hat{z}$ is given by

$\frac{\partial\bf v}{\partial t}+2({\boldsymbol{\Omega}}\times{\bf
v})+\lambda({\bf v}\cdot{\boldsymbol{\nabla}}){\bf
v}=-\frac{{\boldsymbol{\nabla}}p^{*}}{\rho}+\nu\nabla^{2}{\bf v}+{\bf f},$ (1)

where $p^{*}=p+\frac{1}{2}|{\boldsymbol{\Omega}}\times{\bf v}|^{2}$ is the
effective pressure. We assume ${\boldsymbol{\Omega}}=\Omega\hat{z}$, i.e., the
rotation is about the $z$-axis; see Fig. 1.

Figure 1: Geometry of the rotating fluid. We assume the rotation to be about
the $z$-axis. In this coordinate system, ${\bf k}_{\perp}=(k_{x},k_{y})$ and
$k_{\parallel}=k_{z}$.

In this case (1) may be written in terms of components as

$\displaystyle\frac{\partial v_{z}}{\partial t}+\lambda({\bf
v}\cdot{\boldsymbol{\nabla}})v_{z}=-\frac{\partial_{z}p^{*}}{\rho}+\nu\nabla^{2}v_{z}+f_{z},$
(2) $\displaystyle\frac{\partial v_{x}}{\partial t}-2\Omega v_{y}+\lambda({\bf
v}\cdot{\boldsymbol{\nabla}})v_{x}=-\frac{\partial_{x}p^{*}}{\rho}+\nu\nabla^{2}v_{x}+f_{x},$
(3) $\displaystyle\frac{\partial v_{y}}{\partial t}+2\Omega v_{x}+\lambda({\bf
v}\cdot{\boldsymbol{\nabla}})v_{y}=-\frac{\partial_{y}p^{*}}{\rho}+\nu\nabla^{2}v_{y}+f_{y}.$
(4)

Here, $\lambda=1$. These equations in 3D admit two conserved quantities in the
inviscid limit: (i) kinetic energy $E=\int d^{3}x\rho\,v^{2}/2$ and (ii)
helicity $H=\int d^{3}x\,{\bf v}\cdot{\boldsymbol{\nabla}}\times{\bf v}={\bf
v}\cdot{\boldsymbol{\omega}}$, where
$\boldsymbol{\omega}\equiv{\boldsymbol{\nabla}}\times{\bf v}$ is the local
vorticity. In the viscous steady states, in a Kolmogorov-like picture
neglecting intermittency, $E$ and $H$ should have constant (i.e., scale
independent) fluxes. Clearly, $E/H$ has the dimension of a length, which
allows us to define a length-scale $l^{*}=E/H$. We assume the external forces
to be non-helical, i.e., no helicity injection by the forces. Thus helicity is
generated in the bulk only by the global rotation. We consider the
incompressible limit, i.e., the mass density $\rho=const.$, or equivalently
${\boldsymbol{\nabla}}\cdot{\bf v}=0$. At this stage, it is useful to set up
the notations. Below we use $\tilde{\omega}$ and $\tilde{\Omega}$ to denote
Fourier frequencies, while $\Omega$ and $\boldsymbol{\omega}$ represent the
global rotation frequency and vorticity, respectively.

##  III Scaling analysis

To classify the scaling regimes, we first define the following dimensionless
numbers

(i) Rossby number $R_{o}=U/(2\Omega L)$,

(ii) Reynolds number $R_{e}=\frac{LU}{\nu}$ and

(iii) Ekman number $Ek=R_{o}/R_{e}=\nu/(2\Omega L^{2})$,

where $L$ is the linear system size, and $U$ is a typical velocity. We expect
to find two distinct scaling regimes as characterised by $R_{o}$ (or
$\Omega$):

(i) Weak rotation $\Omega\rightarrow 0$, or $R_{o}\rightarrow\infty$

(iii) Large rotation $\Omega\rightarrow\infty$, or $R_{o}\rightarrow 0$.

Since the rotation picks up a direction (the axis of rotation, here the
$z$-axis), system is generally anisotropic. We therefore construct an
anisotropic scaling theory of the system: we assume scaling under the
transformations

${\bf r}_{\perp}\rightarrow l_{\perp}{\bf r}_{\perp},\,z\rightarrow
l_{\perp}^{\xi}z,\,{\bf v}_{\perp}\rightarrow l_{\perp}^{a_{\perp}}{\bf
v}_{\perp},\,v_{z}\rightarrow l_{\perp}^{a_{z}}v_{z}.$ (5)

Here, ${\bf r}_{\perp}\equiv(x,\,y)$, ${\bf v}_{\perp}\equiv(v_{x},\,v_{y})$.
In a general anisotropic situation, $\xi\neq 1$. We also allow for the
possibility $a_{\perp}\neq a_{z}$, i.e., ${\bf v}_{\perp}$ and $v_{z}$ may not
scale in the same way under spatial rescaling. We further define time-scale
$t$ to scale as

$t\sim l_{\perp}^{\tilde{z}},$ (6)

where $\tilde{z}$ is a dynamic exponent. Furthermore, we define a
phenomenological dimensionless constant $\tilde{\Omega}$ by

$\tilde{\Omega}=\frac{[{\boldsymbol{\Omega}}]}{[{\boldsymbol{\omega}}]},$ (7)

Here, […] implies “in a dimensional sense” jkb-mhd . Clearly, the two limiting
cases $\tilde{\Omega}\rightarrow 0$ and $\tilde{\Omega}\rightarrow\infty$
phenomenologically correspond to $R_{o}\rightarrow\infty$ and
$R_{o}\rightarrow 0$.

We note that by balancing the Coriolis force terms against the advective
nonlinear terms, we can extract a length-scale $L_{o}$. In a scaling sense, we
set

$\Omega v\sim\frac{v^{2}}{L_{0}},$ (8)

giving $v\sim\Omega L_{0}$. Dimensionally speaking, energy dissipation

$\epsilon\sim\frac{v^{3}}{L_{0}}\sim\Omega^{3}L_{0}^{2}.$ (9)

This gives

$L_{0}\sim\sqrt{\frac{\epsilon}{\Omega^{3}}}.$ (10)

The corresponding wavevector $k_{0}\equiv 2\pi/L_{0}$ is the Zeeman
wavevector. For length-scales $L\gg L_{0}$ (or equivalently, for wavevector
$k\ll k_{0}$), we expect Coriolis force terms to be important, effects of the
rotation should be strong, and hence non-K41 spectra should follow. In the
opposite limit of $L\ll L_{0}$ (or $k\gg k_{0}$), Coriolis forces should be
irrelevant, and hence K41 scaling should follow. Thus a dual scaling is
believed to exist 2ds12 ; dual1 ; dual2 ; dual3 . In terms of the
dimensionless numbers, we are interested in $R_{e}\rightarrow\infty$ for fully
developed turbulence. Together with $R_{e}\rightarrow\infty$ (implying fully
developed turbulence) and $Ek\rightarrow 0$ (implying Coriolis forces
dominating over the viscous damping at large scales), we can have two
situations: (i) $R_{o}\rightarrow\infty$ for weak rotation, and (ii)
$R_{o}\rightarrow 0$ for strong rotation. Lastly, one has the dissipation
scale $\eta_{d}$, such that for length scales smaller than $\eta_{d}$, the
dissipation range ensues. Then in terms of the length scales defined above, we
can have the following scenarios. In a sufficiently large system, there should
be adequate scale separations, such that $\eta_{d}\ll L_{0}$, i.e., $L_{0}$
should belong to the inertial range. This should allow for both the scaling
regimes, viz. K41 and non-Kolmogorov scaling regimes to be observed.

###  III.1 Nonrotating isotropic case

For a nonrotating, isotropic fluid, $\Omega=0$ in (3)-(4) gives the usual 3D
isotropic Navier-Stokes equation. Let us briefly revisit the extraction of the
K41 scaling by applying the scaling arguments on the usual 3D Navier-Stokes
equation first. Due to the isotropy of the system, we expect $\chi=1$
strictly, and make no distinction between $l_{\perp}$ and $l_{\parallel}$, the
rescaling factors of ${\bf r}_{\perp}$ and $z$ respectively: $l_{\perp}\sim
l_{\parallel}\sim l$. Demanding scale invariance jkb-mhd , we find

$\frac{\partial{\bf v}}{\partial t}\sim{\bf v}\cdot{\boldsymbol{\nabla}}{\bf
v}\implies l^{a-\tilde{z}}=l^{2a-1}\implies a=1-\tilde{z}.$ (11)

Next, in a mean-field like approach, we assume the kinetic energy flux or the
kinetic energy dissipation per unit mass is scale invariant in the inertial
range. This gives

$\frac{\partial v^{2}}{\partial t}\sim l^{0}\implies 2a=\tilde{z}.$ (12)

Combining then, we get $a=1/3,\,\tilde{z}=2/3$. This corresponds to a 1D
kinetic energy spectra $E(k)\sim k^{-5/3}$, the expected K41 result.

###  III.2 Weak rotation effects: $k\gg 2\pi/L_{0}$

In order to study the effects of weak rotation on the scaling of the energy
spectra, we consider the limit $R_{o}\rightarrow\infty$, or $\Omega\rightarrow
0$. Equivalently, we consider length scales $L\ll L_{0}$, with the
understanding that $L\gg\eta_{r}$, the dissipation scale. In this case, the
Coriolis force is unimportant. Hence $\langle H\rangle\approx 0$, where
$<...>$ implies averages over the statistical steady states. Thus, the flux of
$E$ is the relevant (in the Kolmogorov sense) flux. Then, proceeding as in
Ref. jkb-mhd , we unsurprisingly recover the K41 scaling:

$a_{\perp}=a_{z}=1/3,\,\tilde{z}=2/3,\,\xi=1.$ (13)

The last of the above naturally means isotropic scaling (although geometry
remains anisotropic). Furthermore, if we let $R_{o}\sim l^{\eta}$ and demand
scale invariance of all the terms (including the Coriolis force terms), we
find

$\eta=\tilde{z}=2/3.$ (14)

Thus $R_{o}\rightarrow\infty$ as $l\rightarrow\infty$, which corresponds to
nonrotating and isotropic fully developed turbulence. We have assumed that the
nonlinear coupling constant does not scale under spatial rescaling, which is
consistent with the nonrenormalisation of $\lambda$ due to the Galilean
invariance of the Navier-Stokes equation. The scaling of the viscosity is
controlled by the dynamic exponent $\tilde{z}$.

###  III.3 Strong rotation effects: $k\ll 2\pi/L_{0}$

We next consider the large rotation case, i.e., when $R_{o}\geq{\cal O}(1)$,
or $\Omega\geq{\cal O}(1)$. It is now expected that anisotropy is significant.
Below we analyse the scaling in several equivalent ways.

Balancing different terms of (3) and (4) we find

$\tilde{z}=0,\,a_{\perp}=1,\,a_{z}=\xi.$ (15)

To proceed further, we allow for the possibility that not only the spatial
scaling may be anisotropic, there may be different dynamic exponents for ${\bf
v}_{\perp}$ and $v_{z}$, with $\tilde{z}$ being identified as the dynamic
exponent $\tilde{z}_{\perp}$ of ${\bf v}_{\perp}$. To study this, we
separately consider the contribution to the kinetic energy from the in-plane
velocity ${\bf v}_{\perp}$ and normal component of the velocity $v_{z}$.
Interestingly, the exponents in (15) mean that the flux of the “in-plane
kinetic energy” $E_{\perp}\equiv\int d^{3}x\,v_{\perp}^{2}$ cannot be scale-
independent! Let us now consider the kinetic energy $E_{z}\equiv\int
d^{3}x\,v_{z}^{2}$ flux of $v_{z}$. If we assume $\tilde{z}=0$ is the dynamic
exponent of $v_{z}$ also, then $a_{z}=0=\xi$ can actually keep the flux of
$E_{z}$ scale-independent. However, $a_{z}=0$ is unexpected, as it means
$v_{z}$ does not scale with $l$ at all. Assume the dynamics $z_{\parallel}$ of
$v_{z}$ be non-zero: $z_{\parallel}>0$. Now consider Eq. (2) and balance

$\frac{\partial v_{z}}{\partial t}\sim v_{z}\partial_{z}v_{z}\implies
a_{z}=1-z_{\parallel}.$ (16)

Next, demanding scale-invariance of the flux of $E_{z}$ gives

$2a_{z}=\tilde{z}_{\parallel}\implies
a_{z}=\frac{1}{3},\,\tilde{z}_{\parallel}=\frac{2}{3}.$ (17)

This further means $\xi=a_{z}=1/3$. Notice that with $a_{\perp}=1,\,\xi=1/3$,
$({\bf v}_{\perp}\cdot{\boldsymbol{\nabla}}_{\perp})v_{z}$ and
$v_{z}\partial_{z}v_{z}$ scale the same way. Since we get $a_{\perp}>a_{z}$,
we should have $\langle v_{\perp}^{2}\rangle\gg\langle v_{z}^{2}\rangle$ in
the long wavelength limit, suggesting concentration of the kinetic energy in a
plane normal to the rotation axis gode . This implies an effective two-
dimensionalisation. On the other hand, $\tilde{z}<\tilde{z}_{\parallel}$
implies ${\bf v}_{\perp}\ll v_{z}$ in the long time limit, a conclusion
contradictory to our above inference. In fact, this alternative scenario
implies a type of dimensional reduction, where most of the energy is confined
to the $z$-direction. We are unable to conclusively predict which of these two
scenarios actually holds. Numerical studies should be useful in this regard.

Is the ensuing flow field in the limit $R_{o}\rightarrow\infty$ truly 2D? In
our opinion, the answer is no. First of all, the flow remains overall 3D
incompressible. This means the effective 2D flow field might be 2D
compressible, which is an interesting possibility. Secondly, it is not whether
the direction of the kinetic energy cascade becomes backward, a hallmark of
pure 2D turbulence. Thirdly, enstrophy is a conserved quantity in the inviscid
limit of pure 2D turbulence, whereas it is not expected to be so in the 3D
rotating case even in the limit of high rotation. Therefore, notwithstanding
the dominance of $v_{\perp}$ over $v_{z}$, the resulting flow field should be
fundamentally different from pure 2D nonrotating turbulence. Lastly, since
$z_{\perp}\neq z_{\parallel}$, we find weak dynamic scaling jkb-mhd ; dib . We
now calculate the scaling of the two-dimensional kinetic energy spectra
$E_{\perp}(k_{\perp},k_{\parallel})$ and $E_{z}(k_{\perp},k_{z})$, such that
total kinetic energy $E_{\text{tot}}=\int
dk_{\perp}dk_{z}[E_{\perp}(k_{\perp},k_{z})+E_{z}(k_{\perp},k_{z})]$. The
scaling of $E_{\perp}(k_{\perp},k_{z})$ and $E_{z}(k_{\perp},k_{z})$ can be
obtained as follows. We use the general definition to write in a
dimensional/scaling sense

$\displaystyle v_{m}({\bf k},\tilde{\omega})\sim\int v_{m}({\bf
x},t)\exp(i{\bf k_{\perp}}\cdot{\bf x_{\perp}})$ (18)
$\displaystyle\times\exp(ik_{\parallel}z)\exp(i\tilde{\omega}t)d^{2}x_{\perp}dzdt$
$\displaystyle\sim$ $\displaystyle
l_{\perp}^{a_{m}}l_{\perp}^{2}l_{\parallel}\sim\frac{1}{k_{\perp}^{a_{m}+2}}\frac{1}{k_{\parallel}}.$

Here, $m=\perp,\,\parallel$. Furthermore,

$\displaystyle\langle{\bf v}_{\perp}({\bf k}_{1})\cdot{\bf v}_{\perp}({\bf
k_{2}})\rangle=F_{\perp}(k_{1})\delta({\bf k}_{1}+{\bf k}_{2}),$ (19)
$\displaystyle\langle{v}_{z}({\bf k}_{1}){v}_{z}({\bf
k_{2}})\rangle=F_{\parallel}(k_{1})\delta({\bf k}_{1}+{\bf k}_{2}).$ (20)

Dimensionally then,

$F_{\perp}\sim\frac{1}{k_{\perp}^{a_{\perp}+2}k_{\parallel}},\,F_{\parallel}\sim\frac{1}{k_{\perp}^{a_{\parallel}+2}k_{\parallel}}.$
(21)

This gives for the two-dimensional energy spectra

$E_{\perp}(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-3},\,E_{z}(k_{\perp},k_{z})\sim k_{\perp}^{-5/3}.$ (22)

If we ignore anisotropy, we can define two corresponding one dimensional
energy spectra $E_{\perp}(k)$ and $E_{z}(k)$ from (22) by $E_{\text{tot}}=\int
dk[E_{\perp}(k)+E_{z}(k)]$. Notice that, neglecting anisotropy, the one-
dimensional spectra corresponding to $E_{\perp}(k)\sim k_{\perp}^{-3}$ should
scale as $k^{-2}$ as argued above, in agreement with Refs. dual1 ; dual2 ;
dual3 . Nonetheless, in spite of this agreement, we notice that our results
(22) appear to suggest that $E_{\perp}(k_{\perp},k_{z})$ and
$E_{z}(k_{\perp},k_{z})$ have no $k_{z}$-dependence, which should be
unphysical. We try to rectify this below.

First of all, for large $\Omega$, the scaling should be dominated by the
Coriolis forces. The vorticity ${\boldsymbol{\omega}}$ satisfies 2ds7

$\partial_{t}{\boldsymbol{\omega}}({\bf
k})=-2\Omega\frac{k_{\parallel}\hat{e}_{\bf k}\times{\boldsymbol{\omega}}_{\bf
k}}{k},$ (23)

giving time-scale $\tau\sim k/k_{\parallel}\sim k_{\perp}/k_{\parallel}$ for
$k_{\perp}\gg k_{\parallel}$.

It is thus reasonable to assume that $\tau$ as defined above is the relevant
time-scale when $R_{o}\rightarrow 0$. In what follows below, we do not make
any distinction between $v_{\perp}$ and $v_{z}$. We now impose the scale-
independence of the kinetic energy flux $\Pi$. The energy flux may be
calculated from the Navier-Stokes equations (1). We find

$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle-2\lambda^{2}\int_{{\bf
k,q},\tilde{\omega},\tilde{\Omega}}\bigg{[}M_{imn}({\bf k})M_{ijp}(-{\bf
k})\langle v_{m}({-k+q},-\tilde{\omega}+\tilde{\Omega})v_{p}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})\rangle$ (24) $\displaystyle+$
$\displaystyle M_{jmn}({\bf k})M_{ijp}({\bf q})\langle
v_{i}({-k},-\tilde{\omega})v_{n}({\bf k},\tilde{\omega})\rangle\langle
v_{m}({\bf-k+q},-\tilde{\omega}+\tilde{\Omega})v_{p}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})\rangle$ (25) $\displaystyle+$
$\displaystyle M_{jmn}({\bf k-q})M_{ijp}({\bf k})\langle v_{i}({-\bf
k},-\tilde{\omega})v_{n}({\bf k},\tilde{\omega})\rangle\langle v_{j}({\bf
q},\tilde{\Omega})v_{m}({\bf-q},-\tilde{\Omega})\rangle\bigg{]}.$ (26)

Here, $M_{ijp}({\bf k})=P_{ij}({\bf k})k_{p}+P_{ip}({\bf k})k_{j}$. Since we
are interested in the scaling, it suffices to consider the scaling of $\Pi$,
suppressing indices and wavevector labels. At this one-loop order, suppressing
indices and wavevector labels, and assuming $k_{\perp}\gg k_{\parallel}$

$\Pi\sim\int d^{2}k_{\perp}d^{2}p_{\perp}dk_{\parallel}dp_{\parallel}\int
dt\int d^{2}k_{\perp}\,k_{\perp}^{2}C^{2}\sim const.,$ (27)

where $C\equiv\langle v\,v\rangle$ is the correlation function, again
suppressing indices and wavevector labels. This gives

$C\sim k_{\perp}^{-7/2}k_{\parallel}^{-1/2},$ (28)

where we have used $\int dt\sim\tau$. This implies for the two-dimensional
kinetic energy spectra

$E_{\perp}(k_{\perp},k_{\parallel})\sim Ck_{\perp}\sim
k_{\perp}^{-5/2}k_{\parallel}^{-1/2},$ (29)

as in the wave turbulence theory. It now behooves us to show that the scaling
of $\tau$ with wavevector does not get renormalised at the one-loop order. We
restrict ourselves here to a scaling-level demonstration. As shown in
Appendix, the one-loop self-energy $\Sigma_{ik}({\bf k},\tilde{\omega})$ has
the form

$\displaystyle\Sigma_{ij}({\bf k},\tilde{\omega})\sim\int
d^{3}qd\tilde{\Omega}M_{imn}({\bf k})\langle v_{m}({\bf
q},\tilde{\Omega})v_{r}({-\bf q},-\tilde{\Omega})\rangle$ (30)
$\displaystyle\times$ $\displaystyle G_{ns}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})M_{srj}({\bf k-q}).$

Here, $G_{ns}({\bf k},\tilde{\omega})$ is the propagator defined via

$G_{ns}({\bf k},\tilde{\omega})\equiv\bigg{\langle}\frac{\delta v_{n}({\bf
k},\tilde{\omega})}{\delta f_{s}({\bf k},\tilde{\omega})}\bigg{\rangle}.$ (31)

Considering the one-loop self-energy $\Sigma$ and suppressing indices and
wavevector labels [see also Appendix], we obtain

$\Sigma\sim k_{\perp}^{2}\int d^{2}q_{\perp}dq_{z}\int dtG\,C,$ (32)

where $G$ is a propagator. Assuming the dominant time-scale in the above time-
integral is given by $\tau$, we get (in a scaling sense)

$k_{\perp}^{2}\int
d^{2}q_{\perp}dq_{z}\frac{q_{\perp}}{q_{z}}q_{\perp}^{-7/2}k_{z}^{-1/2}\sim[k_{\perp}]^{3/2}[k_{\parallel}]^{1/2},$
(33)

which is less singular than the the bare form of $\tau$. Thus our scaling
results on the 2D kinetic energy spectra remain unaffected by the advective
nonlinearities in the asymptotic long wavelength limit.

Interestingly, we can also derive the above results by using phenomenological
arguments of suppression of the kinetic energy flux by the helicity generated
by the rotation, which are similar to the arguments set up in Ref. jkb-mhd
for scaling of the energy spectra in the presence of a strong mean magnetic
field in magnetohydrodynamic turbulence. It is known that the predominant role
of a (large) non-zero helicity flux is to hinder the cascade of the kinetic
energy flux kraich . In fact, it is easy to see from (3) and (4) that a large
$\Omega$ should suppress the nonlinear effects, relative to the Coriolis force
terms. Since the nonlinear terms are responsible for the cascade phenomena, we
expect the flux of $E_{\perp}$ to be suppressed by a large $\Omega$. This is
similar to the suppression of the energy flux by a strong mean magnetic field
in fully developed magnetohydrodynamic turbulence jkb-mhd ; see also similar
treatment for turbulence in a stably stratified fluid ab-jkb-bolgiano . We
write

$\bigg{[}\frac{\partial v_{\perp}^{2}}{\partial
t}\bigg{]}\bigg{[}\frac{\omega^{2}}{\Omega^{2}}\bigg{]}\sim l^{0}$ (34)

as the condition of the flux being scale independent. Since dimensionally,
$[\omega^{2}]\sim[v_{\perp}^{2}/l^{2}]$, we get

$v_{\perp}\sim l_{\perp}^{3/4}l_{\parallel}^{-1/4}.$ (35)

This gives

$E_{\perp}(k_{\perp},k_{\parallel})\sim k_{\perp}^{-5/2}k_{\parallel}^{-1/2},$
(36)

in agreement with the conclusion from the wave turbulence theory approaches
2ds7 . It is easy to get the spectra in the opposite limit $k_{\parallel}\gg
k_{\perp}$. In this limit, $\tau\sim k_{\perp}^{0}k_{\parallel}^{0}$. At this
one-loop order, suppressing indices and wavevector labels, and assuming
$k_{\perp}\ll k_{z}$.

$\Pi\sim\int d^{2}k_{\perp}d^{2}p_{\perp}dk_{z}dp_{\parallel}\int
d^{2}k_{\perp}\,k_{z}^{2}C^{2}\sim const.,$ (37)

Proceeding as before, we find

$E_{\perp}(k_{\perp},k_{\parallel})\sim k_{\perp}^{-1}k_{\parallel}^{-2}.$
(38)

In each of these cases, the corresponding one-dimensional spectra, without
making any distinction between $k_{\perp}$ and $k_{\parallel}$ scale as
$k^{-2}$, consistent with the recent shell-model studies on rotating
turbulence new-shell .

A pictorial summary of the scaling regimes are shown in Fig. 2.

Figure 2: Schematic diagram illustrating the scaling regimes at different
length scales. Dissipation range for small scales, K41 scaling regime at the
intermediate scales and rotation dominated scaling regimes at the largest
scales are shown.

In the above, we have implicitly assumed that the kinetic energy flux $E$ is
the relevant flux (in the Kolmogorov sense). As we have discussed above this
holds for length scales $\gg l^{*}$. An interesting case may arise if
$l^{*}\gg L_{0}$, in which case in the window between $L_{0}$ and $l^{*}$,
helicity flux $H$ dominates over $E$. If this indeed holds, the scaling of the
energy spectra might change within this window. We do not discuss this further
here.

##  IV Summary and outlook

In this work we have developed a scaling theory for fully developed
incompressible hydrodynamic turbulence in a rotating fluid in the inertial
range. We have studied the scaling of the energy spectra in the inertial range
for weak and strong rotations, i.e., for small and large Rossby number
$R_{o}$. We argue that for wavevectors smaller than the Zeeman wavevector
rotation is important, whereas in the opposite limit, rotation is unimportant.
It is therefore expected that in the former regime the scaling may be
different from the K41 scaling, but in the other regime K41 scaling should
ensue. The scaling theory that we developed here bears this out.

Our scaling theory reveals that in the rotation dominated regime, not only the
scaling itself is anisotropic (i.e., different dependence on $k_{\perp}$ and
$k_{\parallel}$), the scaling of $v_{\perp}$ and $v_{z}$ are different. Even
the dynamic exponents of $v_{\perp}$ and $v_{z}$ are different, indicating
weak dynamic scaling. This suggests that in the limit of a large rotation, the
flow fields are dominated by only some of the velocity components. However,
our theory cannot conclusively predict whether $v_{\perp}$ or $v_{z}$ will be
the dominant part. Numerical simulations should be useful to make further
conclusions.

We have throughout assumed that the kinetic energy flux is the relevant flux
(in the Kolmogorov sense), neglecting the helicity flux. However, for
sufficiently large rotation, there may be a window of length scales where the
helicity flux is the dominant flux. In this regime, the scaling of the energy
spectra should be different. This will be discussed elsewhere.

##  V Acknowledgement

AB thanks the SERB, DST (India) for partial financial support through the
MATRICS scheme [file no.: MTR/2020/000406].

##  Appendix A Reduction of the kinetic energy flux in rotating turbulence:
perturbation theory

We now calculate the kinetic energy flux $\Pi$ in the perturbation theory. We
derive (26) and calculate $\Pi$ to lowest non-trivial order in $\Omega$. We
start by eliminating pressure from the Navier-Stokes equation (1) in a
rotating frame. We obtain

$\displaystyle(-i\tilde{\omega}+\nu k^{2})v_{i}+2P_{im}({\bf
k})\Omega\epsilon_{mzp}v_{p}({\bf k},\tilde{\omega})$
$\displaystyle+i\frac{\lambda}{2}M_{ijp}({\bf k})\sum_{{\bf
q},\tilde{\omega}}v_{j}({\bf q},\tilde{\omega})v_{p}({\bf
k-q},\,\tilde{\omega}-\tilde{\Omega})=f_{i}.$ (39)

To the lowest order in $\Omega$, we expect $\Pi$ to depend on $\Omega$
quadratically, since the energy cascade should be independent of the sense of
rotation around the $z$-axis, i.e., should be the same for clockwise and
anticlockwise rotations. To this order in $\Omega$, it suffices to expand (39)
to ${\cal O}(\Omega)$ and construct an effective equation:

$\displaystyle(-i\tilde{\omega}+\nu k^{2})v_{i}+\frac{i\lambda}{2}M_{ijp}({\bf
k})\sum_{{\bf q},\tilde{\omega}}v_{j}({\bf q},\tilde{\Omega})v_{p}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})$ (40) $\displaystyle=$ $\displaystyle
f_{i}-2P_{im}({\bf k})\epsilon_{mzp}\Omega\frac{f_{p}}{-i\tilde{\omega}+\nu
k^{2}}.$

Clearly, the last term on the rhs of (40), an effective noise, is the dominant
noise in the hydrodynamic limit. We use (40) to calculate the kinetic energy
flux $\Pi$ to ${\cal O}(\Omega^{2})$. The flux $\Pi$ follows

$\displaystyle\frac{d}{dt}\langle v^{2}\rangle$ $\displaystyle=$
$\displaystyle-\frac{i\lambda}{2}\int_{{\bf
k},\tilde{\omega}}\langle\bigg{[}M_{ijp}({\bf k})\sum_{{\bf
q},\tilde{\omega}}v_{i}({\bf-k},-\tilde{\omega})v_{j}({\bf
q},\tilde{\Omega})v_{p}({\bf k-q},\tilde{\omega}-\tilde{\Omega})$ (41)
$\displaystyle+$ $\displaystyle M_{ijp}({\bf-k})\sum_{{\bf
q},\tilde{\omega}}v_{i}({\bf k},\tilde{\omega})v_{j}({\bf
q},\tilde{\Omega})v_{p}({\bf-
k-q},-\tilde{\omega}-\tilde{\Omega})\bigg{]}\rangle.$

Next we iterate and expand the rhs of (41) up to the one-loop order, which
gives (26) above. Now, to the linear order in $\Omega$,

$\displaystyle\langle v_{j}({\bf
q},\tilde{\Omega})v_{n}({\bf-q},-\tilde{\Omega})\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle f_{j}({\bf q},\tilde{\Omega})f_{n}({-\bf
q},-\tilde{\Omega})\rangle}{\tilde{\Omega}^{2}+\nu^{2}q^{4}}-2P_{js}({\bf
q})\epsilon_{szp}G_{0}^{2}(q,\tilde{\Omega})\Omega\frac{\langle f_{p}({\bf
q},\tilde{\Omega})f_{n}({\bf-q},-\tilde{\Omega})\rangle}{i\tilde{\Omega}+\nu
q^{2}}$ (42) $\displaystyle-$ $\displaystyle 2P_{ns}({\bf
q})\epsilon_{szp}G_{0}^{2}(-q,-\tilde{\Omega})\tilde{\Omega}\frac{\langle
f_{p}({\bf-q},-\tilde{\Omega})f_{j}({\bf
q},\tilde{\Omega})\rangle}{-i\tilde{\Omega}+\nu q^{2}}.$

Substituting in (26) and evaluating, we find $\Pi(\Omega)<\Pi(\Omega=0)$,
indicating reduction of the kinetic energy flux by rotation. Since the
helicity generated by the rotation scales with $\Omega$, we conclude that with
a rising helicity, the kinetic energy flux is suppressed.

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